Sums it up

Let there be two functions $p$ and $q$ such that

• $\displaystyle\sum\limits_{k=1}^{\infty} p(k)$ diverges, while $\displaystyle\sum\limits_{k=1}^{\infty} q(k)$ converges, and

• $f(x) = p(x) + q(x)$ , and $g(x) = p(x)$ .

From these assumptions, we can see that the conditions set above are met, and the resulting summation would converge, as

$\displaystyle\sum\limits_{m=1}^{\infty} (f-g)(m) = \displaystyle\sum\limits_{m=1}^{\infty} [ p(m) + q(m) - p(m) ] = \displaystyle\sum\limits_{m=1}^{\infty} q(m)$ which, from our definition, converges.

As an example, set $p(x) = e^x$ and $q(x) = e^{-x}$ . Then, that will give us $f(x) = e^x + e^{-x}$ , and $g(x) = e^x$ , both known to diverge.