Question

Determine if the following Serie converges or not
\[\sum_{k\in \mathbb{N}}2^{2-k}\]

Answer 1

This?
\[\LARGE \sum_{k\in \mathbb{N}}2^{2-k}\]?

Answer 2

yes exactly

Answer 3

Well, the laws of exponents come into play...
\[\Large 2^{2-k}=2^2\cdot 2^{-k}=\frac4{2^k}\]

Answer 4

Does this ring a bell? :P

Answer 5

yes it does not converges cause its like {4 + 2} + ... until 0 right ? the solution is 6 and it does not converges?

Answer 6

\[\LARGE \sum_{k\in \mathbb{N}}2^{2-k}=\sum_{k=1}^\infty\frac4{2^k}=4\sum_{k=1}^\infty \frac1{2^k}\]

Answer 7

hmm ok the solution is 4 and it does not converges :) ?

Answer 8

You'll note that this is just 4 times an infinite geometric series, and when does a geometric series diverge?

Answer 9

hmm when ?

Answer 10

You don't recall? What is the common ratio of this geometric series?

Answer 11

sorry i have looked internet a little bit but cause of language and weakness at math couldnt come any conclusion.. what i all know is geometric serie possible to solve if q < 1 we can use the formel \[\frac{1}{1-q}\]