One broad theme mathematically moving forward through your classes is that many times a problem becomes simpler when we generalize it. Below we see a couple of slick examples of this.

Example 1

Sometimes by generalizing, we can make an expression familiar.

Problem:

Calculate the sum

\[1 + \frac{2}{7} + \frac{3}{7^2} + \frac{4}{7^3} + \frac{5}{7^4} + \dots + \frac{2020}{7^{2020-1}}\]

To solve this, we generalize by replacing \(1/7\) with \(r\) to get

\[1 + 2r + 3r^2 + 4r^3 + 5r^4 + \dots + 2020r^{2020-1}.\]

We can recognize this as the derivative of the geometric sum

\[1 + r + r^2 + r^3 + r^4 + r^5 + \dots + r^{2020}\]

we know that this sum is \(\frac{1-r^{2021}}{1-r}\), so that

\[\begin{align*} 1 + 2r + 3r^2 + 4r^3 + 5r^4 + \dots + 2020r^{2020-1} & = \frac{d}{dr}\left(\frac{1-r^{2021}}{1-r}\right)\\ & = \frac{2020r^{2021}-2021r^{2020}+1}{(1-r)^2} \end{align*}\]

Now inserting the value of \(r = 1/7\), we get that

\[1 + \frac{2}{7} + \frac{3}{7^2} + \frac{4}{7^3} + \frac{5}{7^4} + \dots + \frac{2020}{7^{2020-1}}\approx 1.361111.\]

Example 2

One of Feynmann’s favorite methods is differentiation under the integral which we will apply here in conjunction with our generalization strategy.

Problem:

Calculate the integral

\[\int xe^x dx.\]

To do this, we can realize this as the special case of \(r = 1\) for the integral

\[\int xe^{rx} dx.\]

By differentiation under the integral

\[\begin{align*} \int xe^{rx} dx & = \int\frac{d}{dr}e^{rx}dx = \frac{d}{dr}\int e^{rx}dx\\ & = \frac{d}{dr}\frac{1}{r}e^{rx} = \left(\frac{x}{r}-\frac{1}{r^2}\right)e^{rx}. \end{align*}\]

Taking \(r = 1\), we find

\[\int xe^x dx = (x-1)e^{rx}.\]