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Let $x_i\in\mathbb{R}$ for all $i\in\mathbb{N}.$ Is the following inequality always true?

$$\left|\sum_{i=0}^{\infty}x_i\right|\leq \sum_{i=0}^{\infty}|x_i|$$

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    $\begingroup$ As long as $\sum_{i=0}^{\infty}|x_i|$ makes sense (series converges absolutely). $\endgroup$ – r9m Dec 28 '14 at 21:35
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    $\begingroup$ @r9m: And even if $\sum_{i=0}^\infty|x_i|$ diverges, it can only diverge to $+\infty$, so the inequality is still trivially true in that case. $\endgroup$ – Bungo Dec 28 '14 at 21:36
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    $\begingroup$ @Bungo: I don't think we can say the inequality is trivially true if the left-hand sum does not exist. $\endgroup$ – André Nicolas Dec 28 '14 at 21:38
  • $\begingroup$ @AndréNicolas: Yep, I mentioned that possibility in the answer below. $\endgroup$ – Bungo Dec 28 '14 at 21:42
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It's true as long as $\sum_{i=0}^\infty x_i$ converges. Otherwise the left hand side may be undefined.

To see this, assume that $\sum_{i=0}^\infty x_i$ converges. By the triangle inequality, for every nonnegative integer $N$ we have $$\left|\sum_{i=0}^{N} x_i\right| \leq \sum_{i=0}^N |x_i| \leq \sum_{i=0}^\infty|x_i|$$ where the rightmost sum may be $+\infty$ (making the inequality trivially true) if the series does not converge absolutely.

Therefore, taking the limit as $N \to \infty$, we have $$\lim_{N \to \infty} \left|\sum_{i=0}^{N} x_i\right| \leq \sum_{i=0}^\infty|x_i|$$ Since $\sum_{i=0}^\infty x_i$ converges and the absolute value is continuous, it is also true that $$\lim_{N \to \infty} \left|\sum_{i=0}^{N} x_i\right| = \left|\sum_{i=0}^\infty x_i\right|$$ so the desired result follows.


Edit to discuss the case where $\sum_{i=0}^{\infty} x_i$ diverges in more detail.

Note the distinction: $\sum_{i=0}^{\infty} |x_i|$ has nonnegative terms, so if it diverges, it can only diverge to $+\infty$. But $\sum_{i=0}^{\infty} x_i$ may diverge by failing to approach any one value (finite or infinite), e.g. $x_i = (-1)^i$. The left hand side doesn't make sense if that happens. The best we can say that if $\sum_{i=0}^{\infty} x_i$ converges or if it diverges to $+\infty$ or $-\infty$, then the inequality is true.

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Yes, it is true, if you interpret $+\infty \le +\infty$ and $x \le +\infty$ for all real $x$.

The triangle inequality for finitely many terms shows inequality for the partial sums of the series. Taking the limit of both sides gives the desired inequality.

One or both series may diverge to $+\infty$, so the interpretation in my first paragraph is necessary.

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