# Evaluate the sum of infinite series.

• meson0731
In summary, the conversation discusses evaluating the sum of a given series and the solution being related to the cosine function, specifically cos(pi/12). The initial attempt at solving the problem by writing out terms was deemed inaccurate and the correct solution was provided by recognizing the series as a Taylor series for the cosine function.
meson0731

## Homework Statement

If possible, evaluate the sum :
http://www4a.wolframalpha.com/Calculate/MSP/MSP31841a0i89gaa1b8f5c80000373e40eh779f93h7?MSPStoreType=image/gif&s=17&w=109&h=47

## The Attempt at a Solution

Not really sure what to do. I've tried writing out the terms but its not a geometric series so it didn't help. The only way I can think of is when writing out the first few terms I get something close the the correct answer, which is 0.9659... the numbers get so close together becuase its an alternating series that only the first few terms really matter... but this obviously isn't an accurate way or correct way to do the problem.

Last edited by a moderator:
It looks an awful lot like a Taylor series for the cosine function to me.

ahh your right lol... its just cos(pi/12). Thanks!

## 1. What is the formula for evaluating the sum of an infinite series?

The formula for evaluating the sum of an infinite series is Sn = a / (1-r), where a is the first term of the series and r is the common ratio.

## 2. How do you determine if an infinite series converges or diverges?

An infinite series converges if the limit of its partial sums approaches a finite value. It diverges if the limit does not exist or approaches infinity.

## 3. Can an infinite series have more than one sum or no sum at all?

Yes, an infinite series can have more than one sum or no sum at all. If the series converges, it will have a unique sum. If it diverges, it may have multiple possible sums or no sum at all.

## 4. Is there a way to evaluate the sum of an infinite series without using the formula?

Yes, there are various methods for evaluating the sum of an infinite series without using the formula. These include using the properties of geometric or telescoping series, using recurrence relations, and using the ratio test or the comparison test.

## 5. How is the sum of an infinite series used in real-life applications?

The sum of an infinite series is used in various real-life applications, such as calculating compound interest, finding the value of infinite geometric shapes, and approximating the area under a curve in calculus. It is also used in fields such as physics, engineering, and finance to model and solve real-world problems.

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