# How do you find a formula for the nth partial sum of the series [5/1*2]+[5/2*3]+[5/3*4]+...+[5/n(n+1)]+... and use it to find the series' sum if the series converges?

To find a formula for the nth partial sum of the series ( \frac{5}{n(n+1)} ), we can first write out a few terms of the series and observe a pattern. Then we can use this pattern to derive a general formula for the nth partial sum.

The series is:

[ S_n = \frac{5}{1 \cdot 2} + \frac{5}{2 \cdot 3} + \frac{5}{3 \cdot 4} + \ldots + \frac{5}{n(n+1)} ]

To find the sum of this series, if it converges, we can use the formula for the nth partial sum and take the limit as ( n ) approaches infinity. If this limit exists, it represents the sum of the series.

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For the identity principle of polynomials:

So:

That gives:

So:

We can notice now that all the terms except the first and the last will erase themselves, so:

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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