# Calculate the sum of the series

• shemer77
In summary, the conversation is about finding the sum of the series (1+2^x)/(3^x) from 1 to infinity. The person encountered difficulty in calculating it and rearranged it to a geometric series with a common ratio of 2/3, which converges towards 2. However, the correct answer is 2.5 and it is not necessary to use any other tests except for the geometric series. They have been stuck on this problem for hours and are looking for further clarification on their approach. Another person suggests including the 1 in the numerator and confirms that the method used is correct.
shemer77
so i am having trouble calculating the sum of the series of
(1+2^x)/(3^x) from 1 to infinity
i rearranged it and made a geometric series where r =2/3 and got that it converges toward 2. however the answer is 2.5? This problem shouldn't need any tests as its it a section of the book that hasant taught any of the tests, except for geometric so what am I doing wrong. I have been stuck on this for hours now? :(

Let's see details for how you went about getting your result.

shemer77 said:
so i am having trouble calculating the sum of the series of
(1+2^x)/(3^x) from 1 to infinity
i rearranged it and made a geometric series where r =2/3 and got that it converges toward 2. however the answer is 2.5? This problem shouldn't need any tests as its it a section of the book that hasant taught any of the tests, except for geometric so what am I doing wrong. I have been stuck on this for hours now? :(

You forgot about the 1 in the numerator:

$$\sum \frac{1 + 2^x}{3^x} = \sum \left(\frac{1}{3^x} + \frac{2^x}{3^x} \right) = \sum \frac{1}{3^x} + \sum \left(\frac{2}{3}\right)^x = ?$$

Borhok has the right method again.
Shemer77, would you let us know if you've gotten the right answer from Bohrok's information or still have further questions? Thanks.

## What is the formula for calculating the sum of a series?

The formula for calculating the sum of a series is: S = a + ar + ar^2 + ... + ar^n-1, where S is the sum, a is the first term, r is the common ratio, and n is the number of terms in the series.

## How do I know if a series is convergent or divergent?

A series is convergent if the sum of its terms approaches a finite value as the number of terms increases. It is divergent if the sum of its terms approaches infinity or negative infinity as the number of terms increases.

## Can I use a calculator to calculate the sum of a series?

Yes, you can use a calculator to calculate the sum of a series. However, it is important to use the correct formula and input the values correctly to get an accurate result.

## What is the difference between an arithmetic and geometric series?

An arithmetic series is a series where the difference between consecutive terms is constant. A geometric series is a series where the ratio between consecutive terms is constant. In other words, in an arithmetic series, the terms increase or decrease by a constant amount, while in a geometric series, the terms increase or decrease by a constant factor.

## How do I find the sum of an infinite series?

To find the sum of an infinite series, you can use the formula S = a / (1-r), where S is the sum, a is the first term, and r is the common ratio. However, this formula only works for geometric series with a common ratio r between -1 and 1. For other types of infinite series, you may need to use other methods such as the ratio test or comparison test to determine if the series is convergent or divergent.

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