Confusion over telescoping sums

Question

Consider the following, taken from the Wikipedia article on telescoping series

It might look pedantic, but I am confused by what is a definition, and what needs proving :

1) is $ \sum_1^{\infty} (a_{n} - a_{n-1}) $ defined to be $ \lim_{n \to \infty} \sum_1^{n} (a_n -a_{n-1}) $ ?

2) if so, do we need to prove that $ \lim_{n \to \infty} \sum_1^{n} (a_n -a_{n-1}) = \lim_{n \to \infty} (a_n - a_0) $ or is that 'obvious' ? For instance do we need to assume actual existence of $\lim_{n \to \infty} (a_n - a_0) $ before making the statement that they are equal ?

Answer 1

Here are the answers to your question:-

1. Yes, $\sum\limits_{n = 1}^{\infty} \left( a_n - a_{n - 1} \right)$ is defined as $\lim\limits_{N \to \infty} \sum\limits_{n = 1}^{N} \left( a_n - a_{n - 1} \right)$. This is because, when we talk about convergence of a series, we are actually talking about the convergence of its partial sums. To understand this, consider how we started learning addition of more than two numbers. If we were given, say, $10$ numbers at a time, then we would take the first two, add them and then add this result to the next number. We would do this procedure till all the given numbers were exhausted.

With series, we do a similar thing. Therefore, we define partial sums. If we want to check if a series $\sum\limits_{n = 1}^{\infty} t_n$ will "give us certain value or not", we look at what happens when we start adding the terms one by one. We define the partial sums as $S_N = \sum\limits_{n = 1}^{N} t_n$, which generates a sequence of real numbers. Now, we know about the convergence and we say that if this sequence of partial sums is convergent, then the series is also convergent.

2. Look at the $N$th partial sum

$$S_N = \sum\limits_{n = 1}^{N} \left( a_n - a_{n - 1} \right) = \left( a_1 - a_0 \right) + \left( a_2 - a_1 \right) + \cdots + \left( a_N - a_{N - 1} \right)$$

Using the associative and commutative properties of real numbers, all terms except $a_N$ and $-a_0$ "get cancelled". Therefore, we get

$$S_N = a_N - a_0$$

Now, it is easy to see that if $\lim\limits_{N \to \infty} a_N = 0$, then $\sum\limits_{n = 1}^{\infty} = \lim\limits_{N \to \infty} S_N = - a_0$.