What is the sum of infinte values of x, where x tends to zero? Is the sum 0, or does it go on to become infinite?

Question

I feel that the sum of all very small numbers should be ultimately infinite, however small they are. But I also feel that since all are tending to zero, even the sum will tend to zero? Which is right, if any?

Answer 1

Both your feeling are wrong: it's known for example that

$$1+\frac1{2^2}+\frac1{3^2}+\cdots+\frac1{n^2}+\cdots=\frac{\pi^2}{6}$$ so this sum is finite not equal to $0$.

Answer 2

You have stumbled on a very large part of basic mathematical analysis: infinite series.

A lot can be said about them, but here is the run-down:

The sum of an infinite number of values is, generally, not defined. However, for a sequence of real numbers $a_1,a_2,\dots$, we can define a formal series

$$a_1+a_2+a_3+\dots = \sum_{n=1}^\infty a_n.$$

We say that this series is convergent with a sum of $S$ if the sequence $a_1, a_1+a_2, a_1+a_2+a_3,...$, i.e. the sequence defined as $S_n = a_1 + a_2 + \dots + a_n$, is convergent with a limit of $S$.

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It can be shown that if $\sum_{n=1}^\infty a_n$, then the limit of $a_n$ must be $0$, but the converse is not true.For example, it is known that $$\sum_{n=1}^\infty \frac1n$$ does not converge (if you sum enough elements, the number gets higher and higher). On the othe hand, the sum $$\sum_{n=1}^\infty \frac 1 {2^n}$$ is convergent, summing to $1$.

Answer 3

If you mean $$\infty\lim_{n\to0^+}n$$ it is undefined. But, if you mean $$\lim_{n\to0^+}{\dfrac1nn}$$ then it is equal to $1$.

Answer 4

Note the following:

$\sum_{k=1}^\infty x = \infty$ as $x\to 0^+$.

$\sum_{k=1}^\infty x = -\infty$ as $x\to 0^-$.

$\sum_{k=1}^\infty x$ has no limit as $x\to 0$.

$\sum_{k=1}^\infty \lim_{x\to 0} x = \sum_{k=1}^\infty 0 = 0$.