Does a Countably Infinite Sum of Zeros Equal Zero?

If you mean the series $0+0+0+\cdots$, its sum is zero. An infinite series is defined through the limits of its finite partial sums, and every partial sum here is exactly zero. The fact that there are infinitely many terms does not make the sum undefined or turn it into a nonzero quantity.

This answer assumes each term being added is the number $0$. If instead you mean a function has countably many zeros, those zeros are locations (roots), not automatically terms of a sum; adding the locations would be a different question.

Why the sum is zero

A series is an ordered list of terms, say $(a_n)_{n=1}^{\infty}$, together with the limit of its partial sums. Its $N$th partial sum is

$$S_N=\sum_{n=1}^{N}a_n.$$

For the all-zero series, $a_n=0$ for every positive integer $n$. Therefore, for every finite $N$,

$$S_N=\sum_{n=1}^{N}0=0.$$

The sequence of partial sums is thus $0,0,0,\ldots$. Its limit is zero, which gives

$$\sum_{n=1}^{\infty}0=\lim_{N\to\infty}S_N=0.$$

The infinite sum is not found by multiplying the value of one term by the number of terms as if infinity were an ordinary integer. It is defined by the limit above. Since every finite partial sum is zero, the limiting value is unambiguous.

A worked example

Consider the series

$$0+0+0+0+\cdots.$$

The first few partial sums are

$$S_1=0,\qquad S_2=0+0=0,\qquad S_3=0+0+0=0.$$

In general, $S_N=0$ for every $N$. To check convergence directly, take any tolerance $\varepsilon>0$. For every $N$, and hence for all sufficiently large $N$,

$$|S_N-0|=0<\varepsilon.$$

That is exactly the definition of $S_N\to 0$. So the series converges, and its sum is zero.

The result also holds if the zero terms are indexed by any countably infinite set rather than by the positive integers. Choose an enumeration of that set; every finite sum in the enumeration is still zero. In this particular case, changing the order cannot affect the answer. More generally, a series with all terms zero is absolutely convergent because $\sum_{n=1}^{\infty}|0|=0$ as well.

What if zeros appear alongside other terms?

A countably infinite number of zero terms contributes nothing to a sum, but it does not determine the sum of the other terms. For example,

$$1+0+\frac12+0+\frac14+0+\cdots=2.$$

The zeros leave the partial sums unchanged at the places where they occur. The nonzero terms determine whether the series converges and, if it does, what its sum is. If a series is not convergent, inserting zero terms does not make it convergent.

For instance, inserting a zero after each term of $1+1+1+\cdots$ still gives partial sums that grow without bound. The added zeros do not cancel the ones.

Common mistakes

• Treating infinity as a number of terms to multiply by zero. The usual definition does not calculate an infinite series as “number of terms times a term.” It takes a limit of finite partial sums.
• Assuming infinitely many terms automatically cause divergence. A series can converge or diverge; its behavior depends on its terms. The all-zero series converges immediately because its partial sums are constant.
• Confusing zero terms with roots of a function. If a function has countably many roots, their number alone says nothing about the sum of the root values. First specify which numbers are being added and in what order or sense.

So, for the standard interpretation—one zero at each countable index—the sum is zero.