Section 5.7

Methods of Summing Series

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Problem 2986

Find the sum of the following series:

11⋅3+13⋅5+15⋅7+⋯ . \frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\cdots .
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Problem 2987

Find the sum of the following series:

11⋅2⋅3+12⋅3⋅4+13⋅4⋅5+⋯ . \frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}+\cdots .
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Problem 2988

Find the sum of the following series:

11⋅2−12⋅3+13⋅4−14⋅5+⋯ . \frac{1}{1 \cdot 2}-\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}-\frac{1}{4 \cdot 5}+\cdots .
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Problem 2989

Find the sum of the following series:

∑n=1∞n(n+1)(n+2)(n+3). \sum _{n=1}^{\infty } \frac{n}{(n+1)(n+2)(n+3)}.
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Problem 2990

Find the sum of the following series:

∑n=1∞1n(n+m)(m is a natural number). \sum _{n=1}^{\infty } \frac{1}{n(n+m)} \quad (m \text{ is a natural number}).
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Problem 2991

Find the sum of the following series:

11⋅2⋅3+13⋅4⋅5+15⋅6⋅7+⋯ . \frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{3 \cdot 4 \cdot 5}+\frac{1}{5 \cdot 6 \cdot 7}+\cdots .
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Problem 2992

Find the sum of the following series:

∑n=2∞1n2−1. \sum _{n=2}^{\infty } \frac{1}{n^{2}-1}.
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Problem 2993

Find the sum of the following series:

∑n=1∞2n+1n2(n+1)2. \sum _{n=1}^{\infty } \frac{2n+1}{n^{2}(n+1)^{2}}.
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Problem 2994

Find the sum of the following series:

∑n=1∞1n(2n+1). \sum _{n=1}^{\infty } \frac{1}{n(2n+1)}.
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Problem 2995

Find the sum of the following series:

∑n=1∞n2n!. \sum _{n=1}^{\infty } \frac{n^{2}}{n!}.
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Problem 2996

Find the sum of the following series:

∑n=0∞2n(n+1)n!. \sum _{n=0}^{\infty } \frac{2^{n}(n+1)}{n!}.
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Problem 2997

Find the sum of the following series:

∑n=1∞1n2(n+1)2. \sum _{n=1}^{\infty } \frac{1}{n^{2}(n+1)^{2}}.
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Problem 2998

Find the sum of the following series:

∑n=1∞1n2(n+1)2(n+2)2. \sum _{n=1}^{\infty } \frac{1}{n^{2}(n+1)^{2}(n+2)^{2}}.
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Problem 2999

Find the sum of the following series:

∑n=0∞(−1)nn(2n+1)!. \sum _{n=0}^{\infty } \frac{(-1)^{n}n}{(2n+1)!}.
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Problem 3000

Find the sum of the following series:

∑n=2∞(−1)nn2+n−2. \sum _{n=2}^{\infty } \frac{(-1)^{n}}{n^{2}+n-2}.
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Problem 3001

Let P(x)=α0+α1x+⋯+αmxmP(x)=\alpha_{0}+\alpha_{1}x+\cdots +\alpha_{m}x^{m}. Find the sum of the series

∑n=0∞P(n)n!xn \sum _{n=0}^{\infty } \frac{P(n)}{n!}x^{n}
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Problem 3002

Find the sum of the following series:

∑n=0∞n2+12nn!xn. \sum _{n=0}^{\infty } \frac{n^{2}+1}{2^{n}n!}x^{n}.
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Problem 3003

Find the sum of the following series:

∑n=0∞(−1)nn3(n+1)!xn. \sum _{n=0}^{\infty } \frac{(-1)^{n}n^{3}}{(n+1)!}x^{n}.
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Problem 3004

Find the sum of the following series:

∑n=0∞(−1)n(2n2+1)(2n)!x2n. \sum _{n=0}^{\infty } \frac{(-1)^{n}(2n^{2}+1)}{(2n)!}x^{2n}.
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Problem 3005

Find the sum of the following series:

∑n=0∞n2xn(2n+1)!. \sum _{n=0}^{\infty } \frac{n^{2}x^{n}}{(2n+1)!}.
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Problem 3006

Using term-by-term differentiation, find the sum of the following series:

∑n=1∞xnn. \sum _{n=1}^{\infty } \frac{x^{n}}{n}.
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Problem 3007

Using term-by-term differentiation, find the sum of the following series:

∑n=1∞(−1)n−1x2nn(2n−1). \sum _{n=1}^{\infty } \frac{(-1)^{n-1}x^{2n}}{n(2n-1)}.
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Problem 3008

Using term-by-term differentiation, find the sum of the following series:

∑n=0∞x4n+14n+1. \sum _{n=0}^{\infty } \frac{x^{4n+1}}{4n+1}.
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Problem 3009

Find the sum of the following series:

∑n=1∞a(a+d)⋯[a+(n−1)d]d⋅2d⋯ndxn(d>0). \sum _{n=1}^{\infty } \frac{a(a+d)\cdots \left[a+(n-1)d\right]}{d \cdot 2d\cdots nd}x^{n} \quad (d>0).
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Problem 3010

Find the sum of the following series:

13(x2)+1⋅43⋅6(x2)2+1⋅4⋅73⋅6⋅9(x2)3+⋯ . \frac{1}{3}\left(\frac{x}{2}\right)+\frac{1 \cdot 4}{3 \cdot 6}\left(\frac{x}{2}\right)^{2}+\frac{1 \cdot 4 \cdot 7}{3 \cdot 6 \cdot 9}\left(\frac{x}{2}\right)^{3}+\cdots .
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Problem 3011

Using term-by-term integration, find the sum of the following series:

∑n=1∞n2xn−1. \sum _{n=1}^{\infty } n^{2}x^{n-1}.
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Problem 3012

Using term-by-term integration, find the sum of the following series:

∑n=1∞n(n+2)xn. \sum _{n=1}^{\infty } n(n+2)x^{n}.
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Problem 3013

Using term-by-term integration, find the sum of the following series:

∑n=0∞(2n+1)x2nn!. \sum _{n=0}^{\infty } \frac{(2n+1)x^{2n}}{n!}.
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Problem 3014

Use Abel's method to find the sum of the following series:

1−14+17−110+⋯ . 1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\cdots .
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Problem 3015

Use Abel's method to find the sum of the following series:

1−13+15−17+⋯ . 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots .
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Problem 3016

Use Abel's method to find the sum of the following series:

1−12+1⋅32⋅4−1⋅3⋅52⋅4⋅6+⋯ . 1-\frac{1}{2}+\frac{1 \cdot 3}{2 \cdot 4}-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}+\cdots .
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Problem 3017

Use Abel's method to find the sum of the following series:

1+12⋅13+1⋅32⋅4⋅15+⋯ . 1+\frac{1}{2} \cdot \frac{1}{3}+\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{1}{5}+\cdots .
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Problem 3018

Find the sum of the following trigonometric series:

∑n=1∞sin⁡nxn. \sum _{n=1}^{\infty } \frac{\sin nx}{n}.
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Problem 3019

Find the sum of the following trigonometric series:

∑n=1∞cos⁡nxn. \sum _{n=1}^{\infty } \frac{\cos nx}{n}.
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Problem 3020

Find the sum of the following trigonometric series:

∑n=1∞sin⁡nαsin⁡nxn. \sum _{n=1}^{\infty } \frac{\sin n\alpha \sin nx}{n}.
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Problem 3021

Find the sum of the following trigonometric series:

∑n=1∞sin⁡2nαsin⁡nxn(0<α<π2). \sum _{n=1}^{\infty } \frac{\sin ^{2} n\alpha \sin nx}{n} \quad \left(0<\alpha <\frac{\pi }{2}\right).
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Problem 3022

Find the sum of the following trigonometric series:

∑n=1∞sin⁡(2n−1)x2n−1. \sum _{n=1}^{\infty } \frac{\sin (2n-1)x}{2n-1}.
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Problem 3023

Find the sum of the following trigonometric series:

∑n=2∞(−1)ncos⁡nxn2−1. \sum _{n=2}^{\infty } (-1)^{n} \frac{\cos nx}{n^{2}-1}.
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Problem 3024

Find the sum of the following trigonometric series:

∑n=1∞cos⁡(2n−1)x(2n−1)2. \sum _{n=1}^{\infty } \frac{\cos (2n-1)x}{(2n-1)^{2}}.
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Problem 3025

Find the sum of the following trigonometric series:

∑n=1∞(−1)n−1sin⁡nxn(n+1). \sum _{n=1}^{\infty } (-1)^{n-1} \frac{\sin nx}{n(n+1)}.
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Problem 3026

Find the sum of the following trigonometric series:

∑n=0∞cos⁡nxn!. \sum _{n=0}^{\infty } \frac{\cos nx}{n!}.
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Problem 3027

Plot the curve

∑n=1∞sin⁡nx⋅sin⁡nyn2=0 \sum _{n=1}^{\infty } \frac{\sin nx \cdot \sin ny}{n^{2}}=0
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Problem 3028

Find the sum of the following series:

∑n=1∞[(n−1)!]2(2n)!(2x)2n. \sum _{n=1}^{\infty } \frac{\left[(n-1)!\right]^{2}}{(2n)!}(2x)^{2n}.
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Problem 3029

Find the sum of the following series:

∑n=0∞(n!)2(2n)!xn. \sum _{n=0}^{\infty } \frac{(n!)^{2}}{(2n)!}x^{n}.
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Problem 3030

Find the sum of the following series:

1!x+1+2!(x+1)(x+2)+3!(x+1)(x+2)(x+3)+⋯ . \frac{1!}{x+1}+\frac{2!}{(x+1)(x+2)}+\frac{3!}{(x+1)(x+2)(x+3)}+\cdots .
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Problem 3031

Discuss the series

a1a2+x+a1a2+x⋅a2a3+x+⋯ \frac{a_{1}}{a_{2}+x}+\frac{a_{1}}{a_{2}+x} \cdot \frac{a_{2}}{a_{3}+x}+\cdots

for its convergence and sum, under the condition x>0x>0, an>0a_{n}>0 (n=1,2,⋯n=1,2,\cdots), given that the series ∑n=1∞1an\displaystyle \sum_{n=1}^{\infty } \frac{1}{a_{n}} diverges.

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Problem 3032

Find the sum of the following series:

x1−x2+x21−x4+x41−x8+⋯ , \frac{x}{1-x^{2}}+\frac{x^{2}}{1-x^{4}}+\frac{x^{4}}{1-x^{8}}+\cdots ,

if (a) ∣x∣<1\left|x\right|<1; (б) ∣x∣>1\left|x\right|>1.

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(a)
(b)
Problem 3033

Find the sum of the following series:

∑n=1∞xn+1(1−xn)(1−xn+1), \sum _{n=1}^{\infty } \frac{x^{n+1}}{(1-x^{n})(1-x^{n+1})},

if (a) ∣x∣<1\left|x\right|<1; (б) ∣x∣>1\left|x\right|>1.

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(a)
(b)