Sums it up

Algebra Level 3

Let \( f\) and \(g\) be transcendental functions, such that

  • \( f(x) \neq - g(x) \)

  • \( \displaystyle\sum\limits_{k=1}^{\infty} f(k) \) diverges

  • \( \displaystyle\sum\limits_{j=1}^{\infty} g(j) \) diverges as well.

Must \( \displaystyle\sum\limits_{m=1}^{\infty} (f-g)(m) \) converge?

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