Exam 11: Infinite Sequences and Series

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit. Select the correct answer.

(Multiple Choice)

4.8/5

(32)

Question 141

Determine whether the geometric series converges or diverges. If it converges, find its sum. $- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots$

(Multiple Choice)

4.9/5

(40)

Question 142

Write the first five terms of the sequence $\left\{ a _ { n } \right\}$ whose $n ^ { \text {th } }$ term is given. $a _ { n } = \frac { n + 7 } { 6 n - 1 }$

(Short Answer)

4.8/5

(42)

Question 143

Determine whether the geometric series converges or diverges. If it converges, find its sum. $- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots$

(Short Answer)

4.9/5

(33)

Question 144

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n + 2 }$

(Short Answer)

4.9/5

(41)

Question 145

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }$ Select the correct answer.

(Multiple Choice)

4.9/5

(41)

Question 146

$\text { Determine which one of the } p \text {-series below is convergent. }$

(Short Answer)

4.9/5

(38)

Question 147

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.

(Multiple Choice)

4.8/5

(43)

Question 148

Approximate the sum to the indicated accuracy. $\sum _ { n = 1 } ^ { \infty } \frac { 4 ( - 1 ) ^ { n - 1 } } { n ^ { 7 } }$ (five decimal places)

(Short Answer)

4.9/5

(35)

Question 149

Find a power series representation for the function. $f ( y ) = \ln \left( \frac { 11 + y } { 11 - y } \right)$

(Short Answer)

4.7/5

(34)

Question 150

Find all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges.

(Multiple Choice)

4.9/5

(40)

Question 151

Suppose that the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$ is 9 . What is the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

(Short Answer)

4.9/5

(38)

Question 152

Find the sum of the ser $\sum _ { n = 0 } ^ { \infty } \frac { 2 ^ { n } } { 3 ^ { n } n ! }$

(Short Answer)

4.7/5

(36)

Question 153

Find the sum of the series. $\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots$

(Multiple Choice)

4.8/5

(33)

Question 154

Use the sum of the first 9 terms to approximate the sum of the following series. $\sum _ { n = 1 } ^ { \infty } \frac { 6 } { n ^ { 7 } + n ^ { 2 } }$ Write your answer to six decimal places.

(Short Answer)

4.8/5

(32)

Question 155

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges find its limit.

(Short Answer)

4.9/5

(42)

Question 156

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

(Short Answer)

4.8/5

(31)

Question 157

Given the series $\sum _ { m = 1 } ^ { \infty } \frac { 3 m } { 4 ^ { m } ( 3 m + 5 ) }$ estimate the error in using the partial sum $s _ { 8 }$ by comparison with the series $\sum _ { m - 9 } ^ { \infty } \frac { 1 } { 4 ^ { m } }$ .

(Short Answer)

4.9/5

(33)

Question 158

Find the value of the limit for the sequence given. $\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}$

(Multiple Choice)

4.8/5

(32)

Question 159

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit. Select the correct answer.

Determine whether the geometric series converges or diverges. If it converges, find its sum. $- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots$

Write the first five terms of the sequence $\left\{ a _ { n } \right\}$ whose $n ^ { \text {th } }$ term is given. $a _ { n } = \frac { n + 7 } { 6 n - 1 }$

Determine whether the geometric series converges or diverges. If it converges, find its sum. $- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots$

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n + 2 }$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }$ Select the correct answer.

$\text { Determine which one of the } p \text {-series below is convergent. }$

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.

Approximate the sum to the indicated accuracy. $\sum _ { n = 1 } ^ { \infty } \frac { 4 ( - 1 ) ^ { n - 1 } } { n ^ { 7 } }$ (five decimal places)

Find a power series representation for the function. $f ( y ) = \ln \left( \frac { 11 + y } { 11 - y } \right)$

Find all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges.

Suppose that the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$ is 9 . What is the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

Find the sum of the ser $\sum _ { n = 0 } ^ { \infty } \frac { 2 ^ { n } } { 3 ^ { n } n ! }$

Find the sum of the series. $\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots$

Use the sum of the first 9 terms to approximate the sum of the following series. $\sum _ { n = 1 } ^ { \infty } \frac { 6 } { n ^ { 7 } + n ^ { 2 } }$ Write your answer to six decimal places.

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges find its limit.

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

Given the series $\sum _ { m = 1 } ^ { \infty } \frac { 3 m } { 4 ^ { m } ( 3 m + 5 ) }$ estimate the error in using the partial sum $s _ { 8 }$ by comparison with the series $\sum _ { m - 9 } ^ { \infty } \frac { 1 } { 4 ^ { m } }$ .

Find the value of the limit for the sequence given. $\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Exam 11: Infinite Sequences and Series

Determine whether the sequence defined by an=n2−56n2+1a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }an​=6n2+1n2−5​ converges or diverges. If it converges, find its limit. Select the correct answer.

Determine whether the geometric series converges or diverges. If it converges, find its sum. −15+125−1125+1625−⋯- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots−51​+251​−1251​+6251​−⋯

Write the first five terms of the sequence {an}\left\{ a _ { n } \right\}{an​} whose nth n ^ { \text {th } }nth term is given. an=n+76n−1a _ { n } = \frac { n + 7 } { 6 n - 1 }an​=6n−1n+7​

Determine whether the geometric series converges or diverges. If it converges, find its sum. −15+125−1125+1625−⋯- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots−51​+251​−1251​+6251​−⋯

Find the radius of convergence and the interval of convergence of the power series. ∑n=0∞xnn+2\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n + 2 }∑n=0∞​n+2xn​

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. ∑n=1∞(−1)narctan⁡nn4\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }∑n=1∞​n4(−1)narctann​ Select the correct answer.

Determine which one of the p-series below is convergent. \text { Determine which one of the } p \text {-series below is convergent. } Determine which one of the p-series below is convergent.

Determine whether the sequence defined by an=n2−56n2+1a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }an​=6n2+1n2−5​ converges or diverges. If it converges, find its limit.

Approximate the sum to the indicated accuracy. ∑n=1∞4(−1)n−1n7\sum _ { n = 1 } ^ { \infty } \frac { 4 ( - 1 ) ^ { n - 1 } } { n ^ { 7 } }∑n=1∞​n74(−1)n−1​ (five decimal places)

Find a power series representation for the function. f(y)=ln⁡(11+y11−y)f ( y ) = \ln \left( \frac { 11 + y } { 11 - y } \right)f(y)=ln(11−y11+y​)

Find all positive values of uuu for which the series ∑m=1∞6um7m\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m∑m=1∞​6um7m converges.

Suppose that the radius of convergence of the power series ∑n=0∞cnxn\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }∑n=0∞​cn​xn is 9 . What is the radius of convergence of the power series ∑n=0∞cnx2n\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }∑n=0∞​cn​x2n .

Find the sum of the ser ∑n=0∞2n3nn!\sum _ { n = 0 } ^ { \infty } \frac { 2 ^ { n } } { 3 ^ { n } n ! }∑n=0∞​3nn!2n​

Find the sum of the series. 21⋅3−222⋅32+233⋅33−244⋅34+…\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots1⋅32​−2⋅3222​+3⋅3323​−4⋅3424​+…

Use the sum of the first 9 terms to approximate the sum of the following series. ∑n=1∞6n7+n2\sum _ { n = 1 } ^ { \infty } \frac { 6 } { n ^ { 7 } + n ^ { 2 } }∑n=1∞​n7+n26​ Write your answer to six decimal places.

Determine whether the sequence defined by an=n2−56n2+1a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }an​=6n2+1n2−5​ converges or diverges. If it converges find its limit.

Use series to evaluate the limit correct to three decimal places. lim⁡x→07x−tan⁡−17xx3\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }limx→0​x37x−tan−17x​ Select the correct answer.

Find the value of the limit for the sequence given. {1⋅9⋅17⋯(7n+1)(7n)2}\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}{(7n)21⋅9⋅17⋯(7n+1)​}

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit. Select the correct answer.

Determine whether the geometric series converges or diverges. If it converges, find its sum. $- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots$

Write the first five terms of the sequence $\left\{ a _ { n } \right\}$ whose $n ^ { \text {th } }$ term is given. $a _ { n } = \frac { n + 7 } { 6 n - 1 }$

Determine whether the geometric series converges or diverges. If it converges, find its sum. $- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots$

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n + 2 }$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }$ Select the correct answer.

$\text { Determine which one of the } p \text {-series below is convergent. }$

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.

Approximate the sum to the indicated accuracy. $\sum _ { n = 1 } ^ { \infty } \frac { 4 ( - 1 ) ^ { n - 1 } } { n ^ { 7 } }$ (five decimal places)

Find a power series representation for the function. $f ( y ) = \ln \left( \frac { 11 + y } { 11 - y } \right)$

Find all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges.

Suppose that the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$ is 9 . What is the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

Find the sum of the ser $\sum _ { n = 0 } ^ { \infty } \frac { 2 ^ { n } } { 3 ^ { n } n ! }$

Find the sum of the series. $\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots$

Use the sum of the first 9 terms to approximate the sum of the following series. $\sum _ { n = 1 } ^ { \infty } \frac { 6 } { n ^ { 7 } + n ^ { 2 } }$ Write your answer to six decimal places.

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges find its limit.

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

Find the value of the limit for the sequence given. $\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}$