limiting sum of series (1 Viewer)

[samuelclarke]

Find the limiting sum of the series:

1/5 + 2/5^2 + 3/5^3 + ......

 

[Carrotsticks]

Set up a triangular array, then sum each one:

1/5 + 1/5^2 + 1/5^3 + 1/5^4 + ...
......+ 1/5^2 + 1/5^3 + 1/5^4 + ...
...................+ 1/5^3 + 1/5^4 + ...

etc etc

The sum of the first row is:



The sum of the second row is:



The sum of the third row is:



And so on.

So if we add all the rows, we acquire the series:



Which in itself is a limiting sum with a = 1/4 and r = 1/5

So using the limiting sum formula one last time, we acquire the answer:

 

[RealiseNothing]

Break it up into:



Now sum each series individually.

The first one is:



Second one:



Third one:



If we keep doing this, you notice that we form another GP?

So the sum of this GP is:



I know there is another way of doing this question, but here's a way I like.

 

[samuelclarke]

thanks again! you're a genius lol