Trying to solve this infinite series?

In summary, the conversation is about trying to solve an infinite series that involves adding fractions with an exponent of n. The person has tried various methods, including using the standard method for solving geometric series, but is still unsure how to proceed. Possible approaches discussed include rewriting the expression and using integration.
  • #1
eNathan
352
2
Trying to solve this infinite series??

Hey folks! I've spent hours trying to solve this and have exhausted all available resources.. I just need to be pointed in the right direction!

Homework Statement


Compute the sum of the infinite series (I believe this is an arithmetico geometric series):
[itex]\sum \frac{n+1}{4^{n}}[/itex]

For n=0..infinity

Homework Equations


I know the standard way to solve a geometric series, but don't know how to solve this type of series.


The Attempt at a Solution


I've set up something like this:
[itex]S_{n} = \sum \frac{n+1}{4^{n}}[/itex]
I've tried multiplying by 1/4, 4 and other logical things, but am just not sure how to proceed.

Thanks in advance!
 
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  • #2
Your expression can be written as
$$\sum_0^\infty \frac{1}{4^{n}} + \sum_1^\infty \frac{1}{4^{n}} + \sum_2^\infty \frac{1}{4^{n}} + \dots$$
(This assumes you start your sum at 0, if you start at 1 you have to modify it a bit.)
 
  • #3
Another approach, perhaps more general in application, is to multiply the terms by xn then integrate. You should then be able to sum the series into closed form and differentiate to get a closed form for the original sum. Finally set x=1.
 

1. How do you solve an infinite series?

Solving an infinite series involves finding a sum or limit of an infinite sequence of numbers. This can be done using various techniques such as the geometric series test, telescoping series, and the ratio test.

2. What is the purpose of solving an infinite series?

The purpose of solving an infinite series is to understand and analyze the behavior of a sequence of numbers that has an infinite number of terms. This can help in calculating values of important mathematical functions and understanding real-world phenomena.

3. Are infinite series always solvable?

No, not all infinite series are solvable. Some series may not have a finite sum or limit, making them divergent. It is important to determine the convergence or divergence of a series before attempting to solve it.

4. Can you use calculus to solve infinite series?

Yes, calculus can be used to solve many types of infinite series. Techniques such as integration, differentiation, and Taylor series can be applied to find the sum or limit of a series.

5. Are there any real-world applications of solving infinite series?

Yes, infinite series have many real-world applications in fields such as physics, engineering, finance, and computer science. For example, they are used in calculating compound interest, modeling physical phenomena, and designing algorithms for data analysis.

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