# Answer Limit of a Series: Sum of 1+(2)^n/(3^n) from 1 to ∞

• fk378
In summary, the conversation discusses determining whether a series is convergent or divergent and finding its sum if it is convergent. The individual approaches of separating the summation into two parts and using the equation 1/(1-r) are discussed, with different answers being obtained. However, it is clarified that the sum of a series is the limit of its partial sums, which were not calculated in the first approach, leading to the discrepancy in answers.
fk378

## Homework Statement

Determine whether the series is convergent or divergent. If convergent, find its sum.

sum of [1+(2)^n] / (3^n)
from 1 to inf

## Homework Equations

I know that the sum of a geometric series is 1/(1-r)

## The Attempt at a Solution

The sum of a series is the limit of its partial sums.
I separate the summation into 2 parts: 1/(3^n) + (2^n)/(3^n)

I can see from this that the limits of both of these approach 0, so I conclude that the sum the series is 0.

However, my book says the answer is 5/2 and I tried to solve this a different way and got 5/2 as well. I re-wrote the separate summations as (1/3)^n + (2/3)^n and notice the ratio, r, is 1/3 and 2/3, respectively. Applying the "relevant equation" of 1/(1-r) I solve the summations and get 5/2.

However, if the sum of a series is the limit of its partial sums, why am I getting a different value for my first attempt?

You are confusing the limit of a sequence with the sum of a series. That's all.

My book says that "the sum of a series is the limit of the sequence of partial sums."

You didn't do any partial sums. You just looked at the limits of the individual terms.

## 1. What is the formula for calculating the sum of 1+(2)^n/(3^n) from 1 to ∞?

The formula for calculating the sum of 1+(2)^n/(3^n) from 1 to ∞ is ∑(1+(2)^n/(3^n)) where n starts at 1 and goes to infinity.

## 2. How do I know if a series has a finite or infinite sum?

A series has a finite sum if the terms in the series approach 0 as n goes to infinity. If the terms do not approach 0, the series has an infinite sum.

## 3. Is it possible for a series to have a negative sum?

Yes, it is possible for a series to have a negative sum if the alternating terms in the series are decreasing and approach 0 as n goes to infinity.

## 4. How can I prove that the sum of 1+(2)^n/(3^n) from 1 to ∞ is finite?

You can prove that the sum of 1+(2)^n/(3^n) from 1 to ∞ is finite by using the limit comparison test. This test compares the given series to a known series with a known sum, and if the two series have similar behavior, then the given series also has the same sum.

## 5. Can the sum of 1+(2)^n/(3^n) from 1 to ∞ be approximated?

Yes, the sum of 1+(2)^n/(3^n) from 1 to ∞ can be approximated by taking the sum of a finite number of terms in the series. As more terms are added, the approximation will become more accurate.

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