Finding the sum of an infinite series

• jspectral
In summary, The given series can be written as a power series, where the coefficient of x^n is 1/(n2^(n+1)). By using the theorem of integrating power series, it can be proven that the series approaches ln(2)/2 as n approaches infinity. The proof can be done by replacing 1/2 with a variable and using the power series representation of ln(1+x).

Homework Statement

$$\sum\frac{1}{n2^(n+1)}$$ from 1 to infinity.

By the way, that 2 is to the power of (n+1), doesn't show clearly.

The Attempt at a Solution

I have worked out the first few individual calculations, up to n=6, and i know it approaches ln(2)/2, however I have no idea how to actually prove this.

What sorts of theorems do you know about power series?

jbunniii said:
What sorts of theorems do you know about power series?

A few, it's for a 2nd year university subject. We've been using proof by induction a fair bit, but other than that I don't know the actual names for them.

What do you know of integrating power series?

TheFurryGoat said:
What do you know of integrating power series?

You mean int(1 + x + x^2 + x^3 + ...) = x + x^2/2 + x^3/3 + ... ?

jspectral said:
You mean int(1 + x + x^2 + x^3 + ...) = x + x^2/2 + x^3/3 + ... ?

Yes. See if you can work out how to apply that property to your problem.

(Hint: start by replacing the "1/2" with a variable.)

By the way, you can click on the following equation if you want to see how to typeset the sum properly:

$$\sum_{n=1}^{\infty}\frac{1}{n 2^{n+1}}$$

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1. What is an infinite series?

An infinite series is a mathematical expression that represents the sum of an infinite number of terms. It is written in the form of ∑(an), where an is the nth term of the series.

2. How do you find the sum of an infinite series?

The sum of an infinite series can be found by using a specific formula or by using a technique called convergence. The formula for finding the sum of a geometric series is S = a1 / (1 - r), where a1 is the first term and r is the common ratio. For other types of series, convergence tests such as the ratio test or the root test can be used to determine if the series converges or diverges. If it converges, the sum can then be found by using various methods such as the integral test, the comparison test, or the limit comparison test.

3. Can all infinite series be summed?

No, not all infinite series can be summed. Some series, known as divergent series, do not have a finite sum and therefore cannot be summed. Examples of divergent series include the harmonic series and the alternating harmonic series. It is important to use convergence tests to determine if a series can be summed.

4. What is the difference between a convergent and a divergent series?

A convergent series is one in which the sum of all the terms is a finite number. In other words, as more terms are added, the sum approaches a specific value. A divergent series, on the other hand, does not have a finite sum and the terms either grow infinitely large or oscillate without settling on a specific value.

5. Why is finding the sum of an infinite series important?

Finding the sum of an infinite series is important in many areas of mathematics, physics, and engineering. It allows for the solution of many problems that involve infinite processes, such as calculating the area under a curve or the value of a continuous function at a certain point. Additionally, understanding infinite series is crucial in understanding more complex mathematical concepts, such as calculus and differential equations.