 Shea Stuart

2022-07-03

Is my reasoning for whether
$F\left(x\right)={\int }_{0}^{x}\sum _{0}^{\mathrm{\infty }}\frac{\mathrm{cos}\left(nt\right)}{{2}^{n}}\text{d}t$ Expert

It looks good, but some additional justifications should be made. At the start (and for the later series) you should state why the series converges uniformly (by the Weierstrass_M-test, e.g.).
Also, to justify that the sum of the series is integrable and that switching the order of summation and integration is valid, you should state that the terms of $G\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{cos}\left(nt\right)}{{2}^{n}}$ are integrable over any interval $\left[0,x\right]$
But, can save a few steps in your argument. Since $\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{cos}\left(nt\right)}{{2}^{n}}$ converges uniformly to $G\left(x\right)$, and since the terms of this series are continuous, $G\left(x\right)$ is a continuous function. The Fundamental Theorem of Calculus immediately gives you the continuity of $F\left(x\right)={\int }_{0}^{x}G\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$

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