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I want to find $\displaystyle \lim_{x \rightarrow 1^-} \sum_{n \in \mathbb N} \frac {(-1)^{n+1}}{n} x^n$

What I'm guessing I'm supposed to do is use a certain property that states that if $\displaystyle \sum_{n \in \mathbb N} \frac {(-1)^{n+1}}{n} x^n$ converges for some $x$ in $[0,1]$ and the series of the derivatives $\displaystyle \sum_{n \in \mathbb N} (-1)^{n+1} x^{n-1}$ converges uniformly in $[0,1]$, then the original series converges uniformly and thus I can exchange the limit on $x$ and the limit on $n$, giving

$$\displaystyle \lim_{x \rightarrow 1^-} \lim_{N \rightarrow \infty} \sum_{n = 1}^N \frac {(-1)^{n+1}}{n} x^n = \lim_{N \rightarrow \infty} \lim_{x \rightarrow 1^-} \sum_{n = 1}^N \frac {(-1)^{n+1}}{n} x^n$$

Which would result in the alternate Harmonic series (multiplied by $-1$).

The problem, however, is that while it is true that the derivatives converge uniformly on $[0,1)$, they don't on $[0,1]$ since when x = 1 the partial sums for the derivatives are $1,0,1,0 \dots$

However, the statement that the derivatives need to converge uniformly on $[0,1]$ is what's making my head spin. The original functions are defined on $[0,1]$ so shouldn't the derivatives be defined on $(0,1)$ ?. If not, what does it mean for the derivatives to converge uniformly on $[0,1]$ ?

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  • $\begingroup$ Do you know measure theory and the dominated convergence theorem? $\endgroup$ Dec 8 '20 at 5:38
  • $\begingroup$ If you want to establish uniform convergence, you can use this. $\endgroup$ Dec 8 '20 at 5:41
  • $\begingroup$ You only need the derivatives to converge uniformly on $x \in [0,a]$ for each $0<a<1$, not $a=1$ (which would be false)... $\endgroup$ Dec 8 '20 at 5:44
  • $\begingroup$ I know it would be false, but the text (reading this from Tao's analysis) says that the derivatives converge uniformly on the closed interval, which is kinda weird $\endgroup$ Dec 8 '20 at 6:00
  • $\begingroup$ Sorry I havn't answered. I've been too focused on studying. I'll check your answer out when I can and tell you if I could follow it $\endgroup$ Dec 12 '20 at 0:04
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Let $f(x)= \sum\limits_{n=1}^{\infty} {(-1)^{n-1} x^{n-1}} $. This is geomertic seris and the sum is $\frac 1 {1+x}$. Now $\int_0^{x} f(t) dt=\sum\limits_{n=1}^{\infty} \frac {(-1)^{n-1} x^{n}} n$. Since $(-1)^{n-1}=(-1)^{n+1}$ the given sum is $\int_0^{x} f(t) dt=\ln (1+t)|_0^{x}=\ln (1+x)$. As $x \to 1$ this tends to $\ln 2$.

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Here is one way forward. We will write the series in terms of the even and odd parts of its summand. Proceeding, we find that

$$\begin{align} \sum_{n=1}^\infty \frac{(-1)^{n-1}x^n}{n}&=\sum_{n=1}^\infty \left(\frac{x^{2n-1}}{2n-1}-\frac{x^{2n}}{2n}\right)\\\\ &=\sum_{n=1}^\infty x^{2n-1}\left(\frac{2n(1-x)+1}{2n(2n-1)}\right)\\\\ &=\sum_{n=1}^\infty \frac{(1-x)x^{2n-1}}{2n-1}+\sum_{n=1}^\infty \frac{x^{2n-1}}{2n(2n-1)}\tag1 \end{align}$$

Now, the second series on the right-hand side of $(1)$ is easily seen to be uniformly convergent for $0\le x\le 1$.

To show that the first series on the right-hand side of $(1)$ converges uniformly on $[0,1]$ we have for any $\varepsilon>0$ (and $N\ge1$)

$$\begin{align} \sum_{n=N+1}^\infty \frac{(1-x)x^{2n-1}}{2n-1}&\le \sum_{n=N+1}^\infty \frac{(1-1/2n)^{2n-1}}{2n(2n-1)}\\\\ &\le \sum_{n=N+1}^\infty \frac{1}{4n(2n-1)}\\\\ &<\frac18\sum_{n=N+1}^\infty \left(\frac1{n-1}-\frac1n\right)\\\\ &=\frac1{8N}\\\\ &<\varepsilon \end{align}$$

whenever $N>\frac1{8\varepsilon}$. Hence, we can assure that

$$\begin{align} \lim_{x\to 1^-}\sum_{n=1}^\infty \frac{(-1)^{n-1}x^n}{n}&=\sum_{n=1}^\infty \frac{1}{2n(2n-1)}\\\\ &=\sum_{n=1}^\infty \left(\frac{1}{2n-1}-\frac1{2n}\right)\\\\ &=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}\\\\ &=\log(2) \end{align}$$

as was to be shown!

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