what is telescopic series?

A telescopic series is a series where each term u_n can be written as the difference of two terms a_n and a_{n+1}. This allows for a lot of the terms to cancel out when you add everything together.

Here's a step-by-step explanation:

1. A telescopic series is a series of the form ∑(u_n), where each term u_n = a_n - a_{n+1} for all n in the set of natural numbers.

2. The name "telescopic" comes from the idea of a telescope, where each individual section extends out but also collapses back into the previous section. In a similar way, many of the terms in a telescopic series "collapse" or cancel out when the series is summed.

3. For example, consider the series ∑(1/n - 1/(n+1)). This is a telescopic series, because each term can be written as a difference of two terms.

4. If we start to write out the terms of this series, we get: 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + ...

5. As you can see, many of the terms cancel out: -1/2 + 1/2, -1/3 + 1/3, -1/4 + 1/4, etc. This leaves us with just the first term of each pair: 1, 1/2, 1/3, 1/4, ...

6. So, the sum of the series is just the sum of these remaining terms. In this case, the series sums to 1.

7. This property of telescopic series - that many of the terms cancel out, making the series easier to sum - is what makes them interesting and useful in mathematics.