The Sum to Infinity of 2n/(n+1)!

The Problem

Find the sum to infinity of 2n/(n+1)!

This infinite series looks like this:

If you’re not familiar with infinite series this problem may seem strange. Essentially it’s saying to imagine that the terms of the series continue indefinitely (notice that each term follows a pattern), then add them all up and see what you get. It seems counterintuitive that if you add up infinitely many numbers the answer could be a finite number, but that is how infinite series sometimes go. Obviously you can’t manually just add them up because there are infinitely many – even if you added together the first thousand, or million, or billion terms you still have infinitely many to go. So we need another approach.

The Solution

For a video explanation of the solution go HERE

The first thing to be aware of in this particular question is the presence of the factorial i.e. ! notation. Factorials work like this:

Notice that the series can be written using summation (Sigma) notation like this:

Then

So, we need to evaluate the sum:

And then multiply our answer by 2.

Some Side Working

For a video explanation of this side working go HERE

We are going to use this result about factorials:

For completeness, let’s look at where that result comes from:

But, notice that

So we have

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Back to the Problem

We can now write

Writing out some terms gives:

Notice that up to the Nth term, the terms cancel (called a telescoping series) leaving just

Using a limit we can then write

Evaluating this limit gives:

So we can finally write:

The Final Result

We can now write

Need to learn more about infinite series?

We have the solution with our Calculus 2 Online Course. Featuring 80 step by step instructional videos and more than 350 relevant practice questions with full solutions. Ideal to support your classroom work, help with homework, and prepare for final exams. Learn more HERE and start your FREE one-week trial.

Any Questions?

Drop us a message if you have a question about any of techniques on this page or about series in general.