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## The Problem

Evaluate the infinite series

If you’re not familiar with evaluating infinite series this problem may seem odd. Essentially it’s saying to imagine that the terms of the series continue indefinitely (according to the rule for each term), then add them all together and see what you get. It may seem counterintuitive that if you add up infinitely many numbers the answer could be a finite, but that’s what sometimes happens with infinite series. Clearly 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 a different approach.

## The Solution

We start the solution by using partial fractions to separate the expression into two fractions.

We can now rewrite the original series definition and start substituting values for n i.e. start writing out some of the terms of the series in the new partial fraction form.

Of course we want to evaluate the sum to infinity not just to the first ‘n’ terms. We can’t do this manually so we use a limit.

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## Using a Limit

To finish off our result we need to use a limit. We don’t have space here to explain the concept of a limit so if you’re unsure about this check for an online resource. Essentially by using a limit, we’re going to imagine what will happen to our series as the n value gets infinitely large. We can write

## The Final Result

We can now write our final result which is