Solve Summation by Parts for Sum[n/3^n]

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Homework Help Overview

The discussion revolves around finding the sum of the series Sum[n/3^n] using the technique of summation by parts. The subject area involves series and sequences, particularly focusing on the manipulation of summations and their convergence properties.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the application of summation by parts, with one attempting to define the sequences involved. Questions arise regarding the definition of the partial sums and the general term for the series. Another participant suggests a connection to geometric series and differentiation techniques.

Discussion Status

The discussion is active, with participants providing insights and questioning the definitions and approaches being used. There is an exploration of different interpretations of the summation technique, and some guidance is offered regarding the relationship to geometric series.

Contextual Notes

Participants note potential discrepancies in the application of the summation by parts formula and seek clarification on the sequence of partial sums. There is an acknowledgment of the need for a clearer understanding of the terms involved in the summation process.

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Homework Statement



Using summation by parts, find Sum[n/3^n].


Homework Equations



Sum[a_k*b_k] = s_n*b_(n+1) - Sum[s_k(b_(k+1)-b_k]


The Attempt at a Solution



Let a_k = 1/3^k and b_k = k. Then b_(k+1)-b_k = 1. But what is s_k? I know that it is 1/3 + 1/3^2 + 1/3^3 + ... but what is the general term? Thanks for your help.
 
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S_n is the sequence of partial sums of a_k. My formula is from Goldberg's Methods of Real Analysis.
 
sum(n*(1/3)^n)
very similar to a geometric series, after one differentiation.

sum(n*(1/3)^n) = (1/3)*sum(n*(1/3)^(n-1))

we know that the sum of a geometric series is 1/(1-q), here q=1/3.

sum = (1/3)*diff(1/(1-(1/3)))

sound familiar?

*it's sum found using the integration/differentiation by parts theorem.
 

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