Geometrical series

Question

 

Find at least 10 partial sums of the series. Does it appear that the series is convergent or divergent?

If convergent, find the sum. If it is divergent, why?

∞
∑ (12) / (-5)^n
n=1

Answer 1

The constant 12 can be taken outside the summation because it applies to every term. The series becomes:

12(-1/5+1/25-1/125+1/625-...). The series is a GP with common factor r=-1/5. the series can be written -(12/5)(1-1/5+1/25-...). Call the series in brackets S. So S=1-1/5+1/25-...+5^-n. Therefore S/5=1/5-1/25+...+5^-(n+1) and S+S/5=1+5^-(n+1). When n is very large 5^-(n+1) becomes very small, and it doesn't matter whether it's positive or negative, so we can say that as n approaches infinity the term becomes 0. Therefore S+S/5=6S/5=1 and S=5/6.

Returning to the original GP, we have -(12/5)S=-12/5*5/6=-2. So the series converges to -2.