infinite series

[missyc8]

the problem: find the exact values for the 1st 4 partial sums, find a closed form for the nth partial sum, and determine whether the series converges by calculating the limit of the nth partial sum. if converges, state the sum.

1) 2+2/5+2/5^2+...+2/5^(5-1)+...

i get s1= 2
s2= 2 + 2/5 = 12/5
s3= 2+ 2/5 + 2/5^2 = 62/25
s4= 312/125
Sn =?? the answer is... (2 ? 2(1/5)^n)/ 1-1/5 = 5/2- 5/2(1/5)^n
lim n->infinity Sn= 5/2, converges

can someone please explain how to find Sn...the teacher decided she was not going to teach this section and that we were to learn it on our own, or with extra help outside of the classroom which i did not have time for last week....thanks!

 

[pka]

\(\displaystyle \left| r \right| < 1\, \Rightarrow \,\sum\limits_{n = 0}^\infty {Ar^n } = \frac{A}{{1 - r}}\)

In this problem \(\displaystyle A=2~\&~r=\frac{1}{5}\).

\(\displaystyle \[\begin{gathered} S = A + Ar + Ar^2 + \cdots Ar^n \hfill \\ rS = Ar + Ar^2 + \cdots Ar^n + Ar^{n + 1} \hfill \\ \left( {1 - r} \right)S = A - Ar^{n + 1} \hfill \\ S = \frac{{A - Ar^{n + 1} }}{{1 - r}} \hfill \\ \end{gathered}\)

Here is the key: \(\displaystyle \left| r \right| < 1\, \Rightarrow \,\lim _{n \to \infty } r^n \to 0\)