without using the formula for infinite geometric series?

kloseyq
2022-01-02
Answered

How do I evaluate the power series

$\sum _{n=1}^{\mathrm{\infty}}\frac{n}{{9}^{n}}$

without using the formula for infinite geometric series?

without using the formula for infinite geometric series?

You can still ask an expert for help

poleglit3

Answered 2022-01-03
Author has **32** answers

Check with induction that the partial sum is

$\sum _{k=1}^{N}\frac{k}{{9}^{k}}=\frac{9-{9}^{-N}(8N+9)}{64}$

Evaluate the limit when N goes to infinity and you get the result.

Evaluate the limit when N goes to infinity and you get the result.

Beverly Smith

Answered 2022-01-04
Author has **42** answers

Notice that $x\frac{d}{dx}\frac{1}{1-x}=\sum _{i=0}^{\mathrm{\infty}}i{x}^{i}$ . So differenciating you get

$x\frac{d}{dx}\frac{1}{1-x}=\frac{x}{{(1-x)}^{2}}$

Substituting$x=\frac{1}{9}$ you get required answer.

Substituting

karton

Answered 2022-01-09
Author has **368** answers

Denote

which gives the value of A. Of course, as a byproduct of the computation one also gets the value of the geometric series C.

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