# Finding the sum of an infinite series

• waealu
In summary, to find the value of the infinite series, you can use the equation cos(n)=((e^i)^n+(e^(-i))^n)/2 and the formula =a/(1-r). However, if you want to solve it using only real numbers, you can consider using angle sum formulas and multiplying by cos(1).
waealu

## Homework Statement

I am supposed to find the value of the infinite series:

$$\sum_{n=0}^{+\infty}\frac{\pi\cos(n)}{5^n}$$

## Homework Equations

I asked this question before on this forum and micromass told me that I should use cos(n)=((e^i)^n+(e^(-i))^n)/2. That equation worked and I was able to find the correct answer. I used that equation to find a "a" and an "r" in order to find the solution using =a/(1-r).

But my instructor said he would like me to try and solve it by only using real numbers.

How would I start to solve that without the imaginary numbers?

Thanks

Last edited:
just an idea, but how about considering angle sum formulas...
ie what is
cos(n)
cos(2n)
cos(3n)
in terms of only n?

try multiplying by cos(1) …

S*cos(1) = ∑ π cos(n)cos(1)/5n = … ?

## What is an infinite series?

An infinite series is a sum of an infinite number of terms, where each term is added to the previous one. It is represented in the form of Σan, where an is the nth term of the series.

## What is the sum of an infinite series?

The sum of an infinite series is the total value obtained after adding all the terms in the series. This value can be finite or infinite, depending on the series and its terms.

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

The sum of an infinite series can be found using various methods such as the geometric series method, telescoping series method, and the ratio test. These methods involve finding patterns in the terms of the series and using mathematical formulas to calculate the sum.

## When can the sum of an infinite series be calculated?

The sum of an infinite series can be calculated when the series is convergent, meaning that the infinite number of terms in the series add up to a finite value. If the series is divergent, the sum cannot be calculated as it either tends to infinity or does not have a well-defined value.

## Why is finding the sum of an infinite series important?

Finding the sum of an infinite series is important in various fields of mathematics, physics, and engineering. It helps in solving real-world problems that involve infinite processes, such as calculating the total distance traveled by an object in an infinite time period. It also plays a crucial role in understanding the behavior and convergence of mathematical functions.

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