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Using complex numbers to evaluate series

Evaluate the following infinite series: $$\sum_{k=0}^\infty \frac{1}{(2k+1)!}$$ $$\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}$$ $$\sum_{k=0}^\infty \frac{1}{(3k)!}$$ $$\sum_{k=0}^\infty \frac{1}{8^k(3k+1)!}$$ $$\sum_{k=0}^\infty \frac{(-1)^k}{3k+1}$$

In this post, I will demonstrate how to evaluate series similar to the the Taylor series of a few common functions. Recall the following ordinary Taylor series: $$e^x=\sum_{k=0}^\infty \frac{x^k}{k!}$$ $$-\ln(1-x)=\sum_{k=1}^\infty \frac{x^k}{k}$$

One notices immediately that these Taylor series look very similar to the series problems that I have proposed, but they only use some of the terms. For example, the series $$\sum_{k=0}^\infty \frac{1}{(2k+1)!}$$ is a special case of $$\sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}$$ which is just the sum of every other term in the Taylor series of $e^x$. How might we eliminate half of the terms to obtain the desired result?

First, I will consider a more general series. Suppose the function $G$ can be expressed as the following series: $$G(x)=\sum_{k=0}^\infty a_k x^k$$ where $a_k$ is some sequence. Then notice that $$G(-x)=\sum_{k=0}^\infty (-1)^k a_k x^k$$ which gives an alternating version of the same series. Thus, if we add $G(x)$ and $G(-x)$, every other term will cancel, and we will have $$G(x)+G(-x)=a_0x^0+a_0x^0+a_1x^1-a_1x^1+a_2x^2+a_2x^2+a_3x^3-a_3x^3+...$$ $$G(x)+G(-x)=2a_0x^0+2a_2x^2+2a_4x^4+...$$ or $$\frac{G(x)+G(-x)}{2}=\sum_{k=0}^\infty a_{2k}x^{2k}$$ Similarly, if we subtract $G(x)$ and $G(-x)$, the even terms cancel and the odd terms remain, giving us $$\frac{G(x)-G(-x)}{2}=\sum_{k=0}^\infty a_{2k+1}x^{2k+1}$$ Great, now we should be able to evaluate our first sum! If we let $G(x)=e^x$, then $a_k=\frac{1}{k!}$, telling us that $$\frac{e^x-e^{-x}}{2}=\sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}$$ and, consequently, $$\frac{e-e^{-1}}{2}=\sum_{k=0}^\infty \frac{1}{(2k+1)!}$$ To evaluate the second sum, we will have to use imaginary numbers and Euler's formula: $$\frac{e^i-e^{-i}}{2}=\sum_{k=0}^\infty \frac{i^{2k+1}}{(2k+1)!}$$ $$\frac{e^i-e^{-i}}{2}=i\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}$$ Now, from Euler's formula, we may say that $e^i=\cos(1)+i\sin(1)$ and $e^{-i}=\cos(1)-i\sin(1)$: $$\frac{\cos(1)+i\sin(1)-(\cos(1)-i\sin(1))}{2}=i\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}$$ $$i\sin(1)=i\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}$$ $$\sin(1)=\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}$$ With the first two sums out of the way, things get a little bit trickier. We've figured out how to skip every other term of a series using its alternating form, but how can we eliminate two out of every three terms to be left with every third term?

To solve this problem, we must venture again into the realm of complex numbers - consider the complex numbers in the form $$\omega_n = \cos \frac{2\pi}{n}+i\sin \frac{2\pi}{n}$$ These numbers have the interesting property that the quantity $$\omega_n^{k}+\omega_n^{2k}+...+\omega_n^{nk}$$ equals $n$ if $k$ is a multiple of $n$ and equals $0$ otherwise (the proof is easy, so I will omit it). This interesting property can be applied to help solve our problem. Let us define $G$ the same as before: $$G(x)=\sum_{k=0}^\infty a_k x^k$$ Now consider the following quantity: $$G(x\omega_n)+G(x\omega_n^2)+...+G(x\omega_n^n)=\sum_{k=0}^\infty a_k x^k (\omega_n^k+\omega_n^{2k}+...+\omega_n^{nk})$$ I already noted that $\omega_n^k+\omega_n^{2k}+...+\omega_n^{nk}$ equals $n$ when $k$ is a multiple of $n$ and zero otherwise, meaning that all terms of the series except for those which are multiples of $n$ will be deleted. Thus, we have $$G(x\omega_n)+G(x\omega_n^2)+...+G(x\omega_n^n)=n\sum_{k=0}^\infty a_{nk} x^{nk}$$ or $$\frac{1}{n}\sum_{k=1}^n G(x\omega_n^k)=\sum_{k=0}^\infty a_{nk} x^{nk}$$ We can use this to evaluate the third sum by letting $a_k=\frac{1}{k!}$, $x=1$, and $G(x)=e^x$. This gives us $$\frac{G(\omega_3)+G(\omega_3^2)+G(\omega_3^3)}{3}=\sum_{k=0}^\infty \frac{1}{(3k)!}$$ Now we may use $$\omega_3=-\frac{1}{2}+\frac{\sqrt 3}{2}i,\space\space \omega_3^2=-\frac{1}{2}-\frac{\sqrt 3}{2}i,\space\space \omega_3^3=1$$ to write $$\frac{e^{-\frac{1}{2}+\frac{\sqrt 3}{2}i}+e^{-\frac{1}{2}-\frac{\sqrt 3}{2}i}+e}{3}=\sum_{k=0}^\infty \frac{1}{(3k)!}$$ $$\frac{e^{-\frac{1}{2}}\big(e^{\frac{\sqrt 3}{2}i}+e^{-\frac{\sqrt 3}{2}i}\big)+e}{3}=\sum_{k=0}^\infty \frac{1}{(3k)!}$$ and by Euler's formula, this is equal to $$\frac{2e^{-\frac{1}{2}}\cos \frac{\sqrt 3}{2}+e}{3}=\sum_{k=0}^\infty \frac{1}{(3k)!}$$ ...which gives us a value for the third sum!

Now let us return to the trick we used before: $$G(x\omega_n)+G(x\omega_n^2)+...+G(x\omega_n^n)=\sum_{k=0}^\infty a_k x^k (\omega_n^k+\omega_n^{2k}+...+\omega_n^{nk})$$ This trick allows us to delete all of the terms other than the $a_{nk}x^{nk}$ terms of the sum, but what if we want the $a_{nk+1}x^{nk+1}$ or $a_{nk+2}x^{nk+2}$ terms instead? The key lies in noticing that $$\omega_n^{-m}G(x\omega_n)+\omega_n^{-2m}G(x\omega_n^2)+...+\omega_n^{-nm}G(x\omega_n^n)=\sum_{k=0}^\infty a_k x^k (\omega_n^{k-m}+\omega_n^{2(k-m)}+...+\omega_n^{n(k-m)})$$ which deletes all terms except those for which $k-m$ is a multiple of $n$, or the $nk+m$ terms. Thus we have $$\omega_n^{-m}G(x\omega_n)+\omega_n^{-2m}G(x\omega_n^2)+...+\omega_n^{-nm}G(x\omega_n^n)=n\sum_{k=0}^\infty a_{nk+m} x^{nk+m}$$ or, since $\omega_n^{-1}=\overline{\omega_n}$, $$\overline{\omega_n}^{m}G(x\omega_n)+\overline{\omega_n}^{2m}G(x\omega_n^2)+...+\overline{\omega_n}^{nm}G(x\omega_n^n)=n\sum_{k=0}^\infty a_{nk+m} x^{nk+m}$$ or, more compactly written, $$\frac{1}{n}\sum_{k=1}^n \overline{\omega_n}^{km}G(x\omega_n^k)=\sum_{k=0}^\infty a_{nk+m} x^{nk+m}$$ Now we may calculate the fourth sum by letting $a_k=\frac{1}{k!}$, $x=\frac{1}{2}$, $G(x)=e^x$, and $m=1$: $$\frac{1}{3}\bigg(\big(-\frac{1}{2}-\frac{\sqrt 3}{2}i\big)e^{-\frac{1}{4}+\frac{\sqrt 3}{4}i}+\big(-\frac{1}{2}+\frac{\sqrt 3}{2}i\big)e^{-\frac{1}{4}-\frac{\sqrt 3}{4}i}+e^\frac{1}{2}\bigg)=\sum_{k=0}^\infty \frac{1}{2^{3k+1}(3k+1)!}$$ $$\frac{2}{3}\bigg(-\frac{1}{2}e^{-\frac{1}{4}}\cos \frac{\sqrt 3}{4}+\frac{\sqrt 3}{2}e^{-\frac{1}{4}}\sin \frac{\sqrt 3}{4}+\sqrt e\bigg)=\sum_{k=0}^\infty \frac{1}{8^{k}(3k+1)!}$$ Yuck!

To evaluate the final sum, we will recall the Taylor series for the natural logarithm: $$-\ln(1-x)=\sum_{k=1}^\infty \frac{x^k}{k}$$ In this case, we will let $G(x)=-\ln(1-x)$, $x=-1$, $a_k=\frac{1}{k}$, $n=3$, and $m=1$. This gives us $$\frac{-\ln(1-x)-\big(-\frac{1}{2}-\frac{\sqrt 3}{2}i\big)\ln\big(1-\big(-\frac{1}{2}+\frac{\sqrt 3}{2}i\big)x\big)-\big(-\frac{1}{2}+\frac{\sqrt 3}{2}i\big)\ln\big(1-\big(-\frac{1}{2}-\frac{\sqrt 3}{2}i\big)x\big)}{3}=\sum_{k=0}^\infty \frac{x^{3k+1}}{3k+1}$$ and $$\frac{-\ln(2)-\big(-\frac{1}{2}-\frac{\sqrt 3}{2}i\big)\ln\big(\frac{1}{2}+\frac{\sqrt 3}{2}i\big)-\big(-\frac{1}{2}+\frac{\sqrt 3}{2}i\big)\ln\big(\frac{1}{2}-\frac{\sqrt 3}{2}i\big)}{3}=\sum_{k=0}^\infty \frac{(-1)^{3k+1}}{3k+1}$$ $$\frac{-\ln(2)+\frac{1}{2}\ln\big(\frac{1}{4}+\frac{3}{4}\big)+\frac{\sqrt 3}{2}i\ln\bigg(\frac{1+\sqrt{3}i}{1-\sqrt{3}i}\bigg)}{3}=\sum_{k=0}^\infty \frac{(-1)^{k+1}}{3k+1}$$ $$\frac{-\ln(2)-\sqrt{3}\arctan \sqrt 3}{3}=\sum_{k=0}^\infty \frac{(-1)^{k+1}}{3k+1}$$ $$\frac{-\ln(2)-\sqrt{3}\frac{\pi}{3}}{3}=\sum_{k=0}^\infty \frac{(-1)^{k+1}}{3k+1}$$ $$\frac{-3\ln(2)-\pi\sqrt 3}{9}=\sum_{k=0}^\infty \frac{(-1)^{k+1}}{3k+1}$$ $$\frac{3\ln(2)+\pi\sqrt 3}{9}=\sum_{k=0}^\infty \frac{(-1)^{k}}{3k+1}$$

Before I end this post, I would like to quickly demonstrate another method that can be used to evaluate sums involving trigonometric functions by using Euler's formula. Again, suppose that we define a function $G$ as $$G(x)=\sum_{k=0}^\infty a_k x^k$$ Then we have $$G(x\cdot e^{i\theta})=\sum_{k=0}^\infty a_k x^k\cdot e^{ik\theta}$$ or, by Euler's formula, $$G(x\cdot e^{i\theta})=\sum_{k=0}^\infty a_k x^k(\cos(k\theta)+i\sin(k\theta))$$ Implying that $$\Re[G(x\cdot e^{i\theta})]=\sum_{k=0}^\infty a_k x^k\cos(k\theta)$$ and $$\Im[G(x\cdot e^{i\theta})]=\sum_{k=0}^\infty a_k x^k\sin(k\theta)$$ This is easy to show, but can result in some interesting sums. For example, the following sum can be proven using the former of these two formulae: $$\sum_{k=0}^\infty \frac{\cos(k\theta)}{k!}=e^{\cos\theta}\cos(\sin\theta)$$

That concludes this post!