Hi, just wondering if anyone can help me with a quick infinite series question:

Show that this series diverges: $\displaystyle \sum_{n=1}^{\infty }\frac{2}{n}-\frac{1}{2^n}$

Thanks :)

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- Aug 5th 2012, 04:59 AMDragonkillerInfinite series
Hi, just wondering if anyone can help me with a quick infinite series question:

Show that this series diverges: $\displaystyle \sum_{n=1}^{\infty }\frac{2}{n}-\frac{1}{2^n}$

Thanks :) - Aug 5th 2012, 05:04 AMProve ItRe: Infinite series
- Aug 5th 2012, 06:53 AMHallsofIvyRe: Infinite series
You know, or should know, that $\displaystyle \sum \frac{1}{2^n}$ converges- it is a geometric series and there is a simple formula for its sum.

IF the given sum converged, then we could write $\displaystyle 2\sum\frac{1}{n}= \sum\left(\frac{2}{n}- \frac{1}{2^n}\right)+ \sum \frac{1}{2^n}$ but, as Prove It said, the series on the left does not converge.