I'm a bit confused as to this problem:
Consider the infinite series:
$$\left(1 - \frac 12\right) + \left(\frac 12 - \frac 13\right) + \left(\frac 13 - \frac 14\right) \cdots$$
a) Find the sum $S_n$ of the first $n$ terms.
b) Find the sum of this infinite series.
I can't get past part a) - or rather I should say I'm not sure how to do it anymore.
Because the problem asks for a sum of the infinite series, I'm assuming the series must be Geometric, so I tried to find the common ratio based on the formula
$$r = S_n / S_{n-1} = \frac{\frac 12 - \frac 13}{1 - \frac 12} = \frac 13$$
Which is fine, but when I check the ratio of the 3rd and the 2nd terms:
$$r = \frac{\frac 13 - \frac 14}{\frac 12 - \frac 13} =\frac 12$$
So the ratio isn't constant... I tried finding a common difference instead, but the difference between 2 consecutive terms wasn't constant either.
I feel like I must be doing something wrong or otherwise missing something, because looking at the problem, I notice that the terms given have a pattern:
$$\left(1 - \frac 12\right) + \left(\frac 12 - \frac 13\right) + \left(\frac 13 - \frac 14\right) + \cdots + \left(\frac 1n - \frac{1}{n+1}\right)$$
Which is reminiscent of the textbook's proof of the equation that yields the sum of the first $n$ terms of a geometric sequence, where every term besides $a_1$ and $a_n$ cancel out and yield
$$a_1 \times \frac{1 - r^n}{1 - r}.$$
But I don't know really know how to proceed at this point, since I can't find a common ratio or difference.
Answer
$$\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\left(\frac13-\frac14\right) +\cdots +\left(\frac{1}{n} - \frac{1}{n+1}\right) $$
$$ = 1-\frac12+\frac12-\frac13+\frac13-\frac14+\frac14 -\cdots +\frac{1}{n} - \frac{1}{n+1}$$
$$ = 1-\left(\frac12-\frac12\right)-\left(\frac13-\frac13\right)-\left(\frac14-\frac14\right)-\space \cdots \space - \left(\frac{1}{n}-\frac{1}{n}\right)-\frac{1}{n+1}$$
Notice how each of the terms in parentheses is zero, so we are left with: $$\boxed{\text{Sum of first n terms: }1-\frac{1}{n+1}}$$
If we want the infinite sum we must take the limit as $n \to \infty$ because $n$ is the number of terms in the sequence. So as $n$ becomes arbitrarily large, $\dfrac{1}{n}$ tends towards $0$ so we get that the sequence of finite sums approaches: $$1-0 = \boxed{1}$$
Not all infinite series need to be arithmetic or geometric! This special one is called a telescoping series.
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