Question

convergence of k(1/3)^k

Answer 1

i actually just figured out

Answer 2

Okay great :)

Answer 3

its geometric and bc r < 1 it converges right?

Answer 4

how do you know it is geometric with a \(k\) attached to it ?

Answer 5

\(\sum\limits_{k=0}^{\infty}(1/3)^k\) is easy to spot as geometric series

but how do you know \(\sum\limits_{k=0}^{\infty} \color{Red}{k}(1/3)^k\) is related to geometric series ?

Answer 6

i guess i don't?

Answer 7

It is easy to see the series converges by rewritign it like this :
\[\sum\limits_{k=0}^{\infty} k(1/3)^k = \sum\limits_{k=0}^{\infty} \dfrac{k}{3^k}\]

Notice the numerator is a polynomial and the denominator is an exponential function which grows wildly. You may use ratio test to conclude that the series converges

Answer 8

yeah that makes a lot of sense i could use some more help ill post a different question in a second!

Answer 9

Sure :) there is a neat way to evaluate this sum if you're interested..

Answer 10

\[\sum\limits_{k=0}^{\infty} k(1/3)^{k}\]
is same as\[\sum\limits_{k=0}^{\infty} nx^n\]
where \(x = 1/3\)

Answer 11

and you know the geometric series
\[\frac{1}{1-x} = \sum\limits_{k=0}^{\infty} x^n\]

differentiate both sides and multiply by \(x\)

Answer 12

i see why you're a 'honorary professor of mathematics'