Absolute value of a summation?

Question

What is $|\sum_{k=1}^\infty{f(k)}|$ equal to? Is it just $\sum_{k=1}^\infty|f(k)|$, or something completely different?

Answer 1

Those two are completely different in some cases. For alternating series, this difference becomes apparent, but with other completely positive/negative series, there is no difference.

You can consider for example the Alternating Harmonic series.

$$\sum_{k=1}^\infty\frac{(-1)^k}{k}$$

This converges (which you can prove with the alternating series test). But if you take the absolute value of each term, then it diverges (a well-known result), giving something completely different.

In fact, these two results are so different that there is a term called "Conditional Convergence" to describe how the value of some infinite series diverges if $\sum_{k=1}^\infty|f(x)|$ is considered, but converges if $\sum_{k=1}^\infty f(x)$ is considered.

Answer 2

It depends.
Case 1: If all $f(k)$ have the same sign (i.e., all are either nonnegative or nonpositive), then what you say is correct.
Case 2: If there are two nonzero terms with opposing signs, then you don't have the same term (unless both series diverge).

Answer 3

Well the triangle inequality yields that $$ \left|\sum_{k=1}^\infty{f(k)}\right|\leq \sum_{k=1}^\infty |f(k)|. $$