# Closed Form for nth Partial Sum of a Geometric Series

• Mdhiggenz
In summary, the conversation discusses finding the closed form for the nth partial sum of a geometric sequence and determining if the series converges. The correct equation for the partial sum is sn = 2.5 + 0.25n and the limit of the nth partial sum as n approaches infinity is infinity, indicating that the series diverges.
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+...2/5k-1

## The Attempt at a Solution

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

with R = .2

sn=2+2/5+2/25+2/5n-1

.2sn= 2/5+2/25+2/125+2/5n

I subtract the equations giving me

.8sn=2+2/5n

I divide .8 to both sides giving me

sn= 5/2+1/2n

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

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!

## 1. What is the formula for finding the nth partial sum of a geometric series?

The formula for finding the nth partial sum of a geometric series is Sn = a(r^n - 1)/(r - 1), where a is the first term in the series, r is the common ratio, and n is the number of terms in the series.

## 2. How is the formula for the nth partial sum of a geometric series derived?

The formula for the nth partial sum of a geometric series is derived using the formula for the sum of a finite geometric series, Sn = a(1 - r^n)/(1 - r). By taking the limit as n approaches infinity, we can derive the closed form formula for the nth partial sum.

## 3. What is a geometric series and how is it different from an arithmetic series?

A geometric series is a series where each term is multiplied by a common ratio to get the next term. In an arithmetic series, each term is added by a constant value to get the next term. In a geometric series, the terms can either increase or decrease, while in an arithmetic series, the terms always increase.

## 4. Can the formula for the nth partial sum of a geometric series be used for an infinite series?

No, the formula for the nth partial sum of a geometric series can only be used for a finite series. To find the sum of an infinite geometric series, we use the formula S = a/(1 - r), where a is the first term and r is the common ratio.

## 5. How can the formula for the nth partial sum of a geometric series be used in real-world applications?

The formula for the nth partial sum of a geometric series can be used in finance to calculate compound interest, in physics to calculate the position of an object under constant acceleration, and in computer science to calculate the number of iterations in a loop with a geometrically changing condition.

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