Closed Form for nth Partial Sum of a Geometric Series

[Mdhiggenz]

Homework Statement

Find a closed form for the nth partial sum, and determine whether the series converges by calculating the limit of the nth partial sum.

1. 2+2/5 + 2/25+...2/5^k-1

Homework Equations

The Attempt at a Solution

What I did was I found out it was a geometric sequence.

with R = .2

s_n=2+2/5+2/25+2/5^n-1

.2s_n= 2/5+2/25+2/125+2/5ⁿ


I subtract the equations giving me

.8sⁿ=2+2/5_n

I divide .8 to both sides giving me

s_n= 5/2+1/2ⁿ

I know I am messing up with the multiplication / subtraction of the Nth term much help is appreciated (:

 

[mighty2000]

Hi there,

Great job on identifying the series as a geometric sequence! Your approach to finding the closed form of the nth partial sum is correct, but there is a small error in your calculation. When you subtract the two equations, the left side should be .8sn, not .8sn^2. So the correct equation should be:

.8sn = 2 + 2/5n

When you divide both sides by .8, you should get:

sn = 2.5 + 0.25n

This is the closed form for the nth partial sum. To determine if the series converges, we can take the limit of the nth partial sum as n approaches infinity. In this case, the limit would be infinity, which means the series diverges. This makes sense because as n gets larger and larger, the partial sum also gets larger and larger, indicating that the series does not have a finite sum.

I hope this helps clarify your approach and solution. Keep up the good work!