Pre-interview_Test_Question_39

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so firstly

\begin{align} \int_0^1 \frac 1 {1+x} dx &= [\ln(1+x)]_0^1 \\ &= \ln(1+1)-\ln(1+0) \\ &= \ln2 \end{align}

now we know that \dfrac 1 {1+x}=\dfrac 1 {1 - (-x)}=\displaystyle\sum_{n=0}^{\infty}(-1)^nx^n as |x|<1 (remember x will only range between the limits of the integral), by the formula for the infinite sum of a geometric progression.
substituting this into the integral, we have:

\begin{align} \int_0^1 \frac 1 {1+x} dx &= \int_0^1 \sum_{n=0}^{\infty}(-1)^nx^n dx \\ &= \sum_{n=0}^{\infty} \int_0^1 (-1)^nx^n dx \ \ \ \ (*) \\ &= \sum_{n=0}^{\infty}(-1)^n \int_0^1 x^n dx \\ &= \sum_{n=0}^{\infty} (-1)^n \left[\frac{x^{n+1}}{n+1} \right]_0^1 dx \\ &= \sum_{n=0}^{\infty} (-1)^n \left(\frac 1 {n+1} -0\right) dx \\ &= \sum_{n=0}^{\infty}\frac{(-1)^n}{n+1} dx \\ &= 1-\frac 1 2 + \frac 1 3 - \frac 1 4 + \ldots \end{align}

\therefore we can conclude \ln2 = 1-\dfrac 1 2 + \dfrac 1 3 - \dfrac 1 4 + \ldots
(*) you can interchange the integral and sum as long as both the sum and resulting integral converge.

fwiw convergence is not quite sufficient for the interchange. Uniform convergence [and integrability of each individual term in the sum but this is “obvious” since they’re all continuous] however is, so if you wanted to justify it rigorously (obviously beyond the scope of this question), I’d note that since Taylor series converge uniformly on compact (in \mathbb R^n this just means closed and bounded) subsets of their radius of convergence, we have \displaystyle \sum_{n = 0}^\infty (-1)^n x^n converges uniformly on [0,t] for all 0 \le t < 1, so:

\begin{align*}\int_0^t \left(\sum_{n = 0}^\infty (-1)^n x^n\right) \mathrm dx & = \sum_{n = 0}^\infty \left(\int_0^t (-1)^n x^n \mathrm dx\right) \\ & = \sum_{n = 0}^\infty \frac {(-1)^n} {n + 1} t^{n + 1} \\ & = \ln(1 + t)\end{align*}

taking t \to 1^- gives the result. (https://en.wikipedia.org/wiki/Abel's_theorem)

Don’t worry if this is all analytic gobbledegook at this point, just thought I’d chip in! This is the type of thing you’d see in the second year.

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This is really valuable!!

v cool lol xD
i like analysis, proves so many intuitive results rigorously

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