MATH114 Integration and Differentiation

Problem Solving Class 2

[2.5ex]

• Both (definite) integrals and series can be defined as limits of certain finite sums. How are they related or what are the differences?

• Let $f:[1,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a continuous decreasing function. Can you bound ${\int }_1^{n}f(x)dx$ below and above by a nice finite sum? Provide a few useful expressions. A picture could help to start with.

• Invent a “convergence test” for series which uses integrals, i.e., a criterion that tells you whether a given series converges or diverges.

• Determine for which rationals $\alpha >0$ the series ${\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n^{\alpha }}$ converges. Prove your answer, e.g. by using your new convergence test.

• Consider the function $f:ℝ\to ℝ$ defined by $f(x)=\sin (\pi x)$. Does the corresponding series ${\sum }_{n=1}^{\mathrm{\infty }}f(n)$ converge? Does the integral ${\int }_1^{\mathrm{\infty }}f(x)dx$ converge? Does this not contradict your new convergence test?