Find the sim of each of the following series. 1) ∑n=1∞nxn, |x|&lt;1 2) ∑n=1∞n8n

1) Find the sum of the series ∑n=1∞nxn, |x|&lt;1 ∑n=1∞nxn=x+2x2+3x3+4x4+... =x(1+2x+3x2+4x3+...) =x∑n=1∞nxn−1 =x1(1−x)2 =x(1−x)2 2) Find the sum of the series ∑n=1∞n8n ∑n=1∞n8n=∑n=1∞n(18)n Compare with ∑n=1∞nxn, then x=18 ∑n=1∞nxn−1=x(1−x)2 =1/8(1−18)2 =1/8(7/8)2 =1/849/64 =849

"Find the sim of each of the following..." solved and explained

DofotheroU

Answered question

2021-05-23

Find the sim of each of the following series.
1) $\sum_{n = 1}^{\infty} n x^{n}, | x | < 1$
2) $\sum_{n = 1}^{\infty} \frac{n}{8^{n}}$

Answer & Explanation

Nicole Conner

Skilled2021-05-24Added 97 answers

1) Find the sum of the series $\sum_{n = 1}^{\infty} n x^{n}, | x | < 1$
$\sum_{n = 1}^{\infty} n x^{n} = x + 2 x^{2} + 3 x^{3} + 4 x^{4} + . . .$
$= x (1 + 2 x + 3 x^{2} + 4 x^{3} + . . .)$
$= x \sum_{n = 1}^{\infty} n x^{n - 1}$
$= x \frac{1}{(1 - x)^{2}}$
$= \frac{x}{(1 - x)^{2}}$
2) Find the sum of the series $\sum_{n = 1}^{\infty} \frac{n}{8^{n}}$
$\sum_{n = 1}^{\infty} \frac{n}{8^{n}} = \sum_{n = 1}^{\infty} n (\frac{1}{8})^{n}$
Compare with $\sum_{n = 1}^{\infty} n x^{n}$ , then $x = \frac{1}{8}$
$\sum_{n = 1}^{\infty} n x^{n - 1} = \frac{x}{(1 - x)^{2}}$
$= \frac{1 / 8}{(1 - \frac{1}{8})^{2}}$
$= \frac{1 / 8}{(7 / 8)^{2}}$
$= \frac{1 / 8}{49 / 64}$
$= \frac{8}{49}$

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