The sum of (1/x) from 1 to infinity?

Question

What is the sum?
Thanks.

As a series.

Answer 1

The sum diverges. We can show this with a comparison test.

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + ...

= 1/1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + ...

> 1/1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + ...

= 1 + 1/2 + 1/2 + 1/2 + ... diverges

So the original series diverges also.

Answer 2

That's called the harmonic series, which diverges.

There are several different ways to prove this. One way is to think of this as the sum of the areas of rectangles whose width is 1 and whose height is 1/n. If you were to line these rectangles up along the x axis (so that the first rectangle is bounded by the x-axis, x=1, x=2 and y=1/2, and the second was bounded by x=2, x=3 and y=1/3 etc., creating increasingly small "steps"), then the curve y = 1/x would touch the top left corners of these steps. The area of all these rectangles put together is certainly more than the area under the curve of y = 1/x from 1 to infinity. But integrating 1/x gives Ln(x). This certainly diverges if you tried to put in the range 1 to infinity. So the harmonic series must diverge also.

The harmonic series is a good example of how a series can diverge even if the limits of the individual terms go to zero.

Answer 3

as it goes to infinity, the answer will also go to infinity, it will just get there slower but infinity is infinity