How do I evaluate the power series \sum_{n=1}^\infty\frac{n}{9^n} without using the formula

kloseyq 2022-01-02 Answered
How do I evaluate the power series
n=1n9n
without using the formula for infinite geometric series?
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poleglit3
Answered 2022-01-03 Author has 32 answers
Check with induction that the partial sum is
k=1Nk9k=99N(8N+9)64
Evaluate the limit when N goes to infinity and you get the result.
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Beverly Smith
Answered 2022-01-04 Author has 42 answers
Notice that xddx11x=i=0ixi. So differenciating you get
xddx11x=x(1x)2
Substituting x=19 you get required answer.
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karton
Answered 2022-01-09 Author has 368 answers

Denote A=n1n9n,B=n2n9n, C=n19n, D=n29n. You have the closed system
A=B+19, 9BA=C, C=D+19, 9D=C
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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