Series question help (1 Viewer)

movie_detective · Jan 12, 2013

Hey again all,

The question is : An infinite geometric series is such that the sum of all the terms after the nth term is equal to twice the nth term. Show that the sum to infinity of the whole series is three times the first term.

Thx For helping

asianese · Jan 13, 2013

movie_detective · Jan 13, 2013

Your helping me heaps asianese , thx. However, just curious as to whether we could use the sum of a geometric series formula instead of the sum to infinity formula to get the same answer?

asianese · Jan 13, 2013

The reality is that the infinite formula for GP is based on the 'original' sum for GPs.

$S_{GP}=\frac{a(1-r^n)}{1-r}$ as you know. Now assume that $|r|<1$ and then let $n\rightarrow \infty$ . Since r is between -1 and 1, if we keep having higher powers of it, it will get smaller and smaller. E.g. $\left(\frac{1}{2}\right)^{50}=8.88\times 10^{-16}$ , practically 0. So we can treat $r^n$ as 0 if we put the limit at the front of the sum. So $\lim_{n \to \infty}S=\lim_{n\to \infty} \frac{a(1-r^n)}{1-r}\\=\frac{a(1-0)}{1-r}\\=\frac{a}{1-r}$ which IS the infinite series formula.

You must note that such two formulae are actually the same thing, except one has special conditions (for an infinite converging series, $|r|<1$ ).

movie_detective · Jan 13, 2013

Nice to know that they are the essentially the same and the reasoning behind it.

Series question help (1 Viewer)

movie_detective

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asianese

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movie_detective

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asianese

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