Examples of the use of infinite series
a. General: Zeno's paradoxes
b. Physics: Using the first order Taylor approximation $\sin \theta \approx \theta$ in solving the pendulum differential equation
c. Chemistry: Extending the ideal gas law to apply to high pressure and low temperature situations
d. Economics: Calculating fiscal multipliers involves geometric series
e. Computer science, 1: Several uses for generating functions (see examples by robinhoode and Raphael)
f. Computer science, 2: Taylor series are involved in the error analysis of some numerical methods, such as Newton-Raphson and Simpson's rule.
g. Mathematics, 1: Taylor series show that calculations involving functions like $e^x$ and $\sin x$ can all be computed using just addition, subtraction, multiplication, and division.
h. Mathematics, 2: Power series, and Euler products in number theory in particular, as most people find number theory intrinsically interesting whether they have the background or not
i. Mathematics, 3: Taylor series can be used to solve differential equations. (Often students will have seen a brief introduction to differential equations earlier in the course.)
j. Mathematics, 4: There are infinite series expressions for interesting constants such as $\pi$ and $e$. Also, any nonterminating decimal representation of a real number is an infinite series.
k. Mathematics, 5: Using Taylor polynomials to approximate integrands in definite integrals. (This fits well in a course like Calculus II that spends a lot of time on the integral.)
Best Answer
Hints.
(a) $\dfrac{1}{n(2n-1)}=\dfrac{a}{n}+\dfrac{b}{2n-1}$.
(b) For $|z|< 1$, the derivative of $\sum_{n\geq 0} z^n=\frac{1}{1-z}$ is $\sum_{n\geq 1} nz^{n-1}=\frac{1}{(1-z)^2}$.
(c) $\dfrac{n^3}{n!}=\dfrac{an(n-1)(n-2)+bn(n-1)+c n}{n!}$.
(d) $\dfrac{1}{1+2+\dots +n}=\dfrac{2}{n(n+1)}=\dfrac{a}{n}+\dfrac{b}{n+1}$.
(e) $\dfrac{n^3+n+3}{n+1}=an^2+bn+c+\dfrac{d}{n+1}$.
Can you take it from here? Now you need to remember some basic power series. For example for (c), recall that for any real $x$, $\sum_{n\geq 0}x^n/n!=e^x$ and, by the above hint, we get $$\sum_{n\geq 0}\frac{n^3}{n!}x^n=ax^3\sum_{n\geq 3}\frac{x^{n-3}}{(n-3)!} +bx^2\sum_{n\geq 2}\frac{x^{n-2}}{(n-2)!}+cx\sum_{n\geq 1}\frac{x^{n-1}}{(n-1)!}=(ax^3+bx^2+cx)e^x. $$