Dr. Xiaozhou Li

Dr. Xiaozhou Li

Mathematics. Strength. Long-Term Mindset. Think Hard. Train Harder. Live Long.

Problem Sets: Infinite Series (无穷级数)

Published:

Last Update: 2026-03-24

  • (Zeno’s paradox) Achilles (A) and a tortoise (T) have a race. T gets a $100$ m head start, but A runs at $10$ m/s while the tortoise only does $0.1$ m/s. When A reaches T’s starting point, T has moved a short distance away. When A reaches that new spot, T has moved a distance ahead, etc. Zeno claimed that A would never catch T.

  • True or False. Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give a counterexample.
    1. If $\displaystyle \lim_{n \to \infty} a_n = 0$, then $\displaystyle \sum a_n$ is convergent.
    2. If $\displaystyle \lim_{n \to \infty} a_n = L$, then $\displaystyle \lim_{n \to \infty} a_{2n+1} = L$.
    3. If a finite number of terms are added to a convergent series, then the new series is still convergent.
    4. If $0 \le a_n \le b_n$, and $\sum b_n$ diverges, then $\displaystyle \sum a_n$ diverges.
    5. $\displaystyle \sum a_n$ is a divergent series, and $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = c > 0$, then the series $\displaystyle \sum b_n$ diverges.
    6. If $\displaystyle \sum (a_{2n-1} + a_{2n})$ converges, then $\displaystyle \sum a_n$ converges.
    7. If $\displaystyle \sum a_n$ converges, then $\displaystyle \sum a_{n+91}$ converges.
    8. $\displaystyle \sum a_n$ converges, then $\displaystyle \sum (-1)^n \frac{a_n}{n}$ converges.
    9. If $\displaystyle \lim_{n \to \infty} \frac{a_{n+1}}{a_n} > 1$, then $\displaystyle \sum a_n$ diverges.

  • Prove that the sequence $\displaystyle {a_n}$ converges if and only if the series $\displaystyle \sum_{n=1}^{\infty} (a_{n+1} - a_n)$ converges.

  • Find the sum of the series
    • $\displaystyle \sum_{n=2}^{\infty} \ln\left(1 - \frac{1}{n^{2}}\right)$
    • $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n(n+m)}$
    • $\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{2n + 1}$
    • $\displaystyle \sum_{n=1}^{\infty} \frac{3n + 5}{3^{n}}$
    • $\displaystyle \sum_{n=0}^{\infty} \arctan\left(\frac{1}{n^{2} + n + 1}\right)$ （Hint: $\displaystyle \arctan\left(\frac{\alpha - \beta}{1 + \alpha\beta}\right) = \arctan\alpha - \arctan\beta$)
    • $\displaystyle \sum_{n=1}^{\infty}\frac{n}{(n+1)!}$
    • $\displaystyle \sum_{n=3}^{\infty} \frac{2^{2 n}-(-11)^{n}}{13^{n}}$
    • $\displaystyle \sum_{k=1}^{\infty} \left( \frac{1}{3k-2} + \frac{1}{3k-1} - \frac{2}{3k} \right)$
    • $\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{12} + \cdots$, where the terms are the reciprocals of the positive integers whose only prime factors are 2s and 3s.

  • If $p > 1$, evaluate the expression \(\displaystyle \frac{1 + \frac{1}{2^{p}} + \frac{1}{3^{p}} + \frac{1}{4^{p}} + \cdots}{1 - \frac{1}{2^{p}} + \frac{1}{3^{p}} - \frac{1}{4^{p}} + \cdots}.\)

  • Prove that if $\displaystyle \lim_{n \to \infty} n a_n = a \neq 0$, then the series $\displaystyle \sum_{n=1}^{\infty} a_n$ diverges.

  • Suppose ${a_n}$  is a sequence of positive terms. Prove that if the $\displaystyle \sum_{n=1}^{\infty} a_n$ converges, then the series $\displaystyle \sum_{n=1}^{\infty} a_n^{2}$ also converges. Provide a counterexample to show that the converse of this statement is not necessarily true.

  • Suppose the series $\displaystyle \sum_{n=1}^{\infty} a_n^{2}$ and $\displaystyle \sum_{n=1}^{\infty} b_n^{2}$ converge, show that the following series also converge: \(\sum_{n=1}^{\infty} \lvert a_n b_n \rvert, \quad \sum_{n=1}^{\infty} (a_n + b_n)^{2}, \quad \sum_{n=1}^{\infty} \frac{\lvert a_n \rvert}{n}.\)

  • Determine if the series is absolutely convergent, conditionally convergent, or divergent.
    • $\displaystyle \sum_{n=5}^{\infty} \frac{2^{2 n}-(-17)^{n}}{13^{n}}$
    • $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$
    • $\displaystyle \sum_{n=2}^{\infty} \frac{1}{(\ln n)^{\ln n}}$
    • $\displaystyle \sum_{n=1}^{\infty} \frac{n^{2}}{\left(1 + \frac{1}{n}\right)^{n}}$
    • $\displaystyle \sum_{n=1}^{\infty} \frac{2^{n} n!}{n^{n}}$
    • $\displaystyle \sum_{n=1}^{\infty} n^{n} e^{-n^{2}}$
    • $\displaystyle \sum_{n=0}^{\infty} \frac{(\ln n)^2}{n!}$
    • $\displaystyle \sum_{n=1}^{\infty} \int_{0}^{\frac{\pi}{n}} \frac{\sin x}{1 + x} \, dx$
    • $\displaystyle \sum_{n=2}^{\infty} (-1)^{n} \left( \sqrt{n^{2} + 1} - \sqrt{n^{2} - 1} \right) n^{p} \ln n\quad p \in [0,1]$.
    • $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^{n}}\left(1 + \frac{1}{n}\right)^{n^{2}}$
    • $\displaystyle \sum_{n=2}^{\infty} \frac{3^{n}}{n \ln (n)}$
    • $\displaystyle \sum_{n=2}^{\infty} n^{2}\tan \frac{\pi}{2^{n}}\sin(n!)$
    • $\displaystyle \sum_{n=1}^{\infty}\left(\cos\frac{1}{\sqrt{n}}\right)^{n^{2}}$
    • $\displaystyle \sum_{n=1}^{\infty} \frac{n !}{(n+3)^{n}}$
    • $\displaystyle \sum_{n=2}^{\infty} \frac{1}{n \ln^{p} n}$
    • $\displaystyle \sum_{n=3}^{\infty} \frac{1}{n (\ln n)^{p} (\ln\ln n)^{q}}$
    • $\displaystyle \sum_{n=2}^{\infty} \sin \left(n\pi + \frac{1}{\ln n}\right)$
    • $\displaystyle\frac{1}{\sqrt{2}+1} - \frac{1}{\sqrt{2}-1} + \frac{1}{\sqrt{3}+1} - \frac{1}{\sqrt{3}-1} + \cdots$
    • $\displaystyle \frac{1}{a} + \frac{1}{a+1} - \frac{1}{a+2} + \frac{1}{a+3} + \frac{1}{a+4} - \frac{1}{a+5} + \frac{1}{a+6} + \frac{1}{a+7} - \frac{1}{a+8} + \cdots$
    • $\displaystyle \sum_{n=2}^{\infty} \frac{(-1)^n}{\sqrt{n+(-1)^n}}$
    • $\displaystyle \sum_{n=2}^{\infty} \frac{\sqrt{n+2} - \sqrt{n-2}}{\sqrt{n}}$
    • $\displaystyle \sum_{n=2}^{\infty} \left(\sqrt[3]{n+1} - \sqrt[3]{n}\right)$
    • $\displaystyle \sum_{n=1}^{\infty}\sin \left(\pi \sqrt{n^{2}+a^{2}}\right)$
    • $\displaystyle \sum_{n=1}^{\infty} \frac{a^{n} n!}{n^{n}}\,\, (a > 0)$
    • $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n - \ln n}$
    • $\displaystyle \sum_{n=1}^{\infty} \left( \sqrt[n]{n} - 1 \right)$
    • $\displaystyle \sum_{n=1}^{\infty} \left( n^{\frac{1}{n^2+1}} - 1 \right)$
    • $\displaystyle \sum_{n=1}^{\infty} \left( \frac{1}{n} - \ln \frac{n+1}{n} \right).$
    • $\displaystyle \sum_{n=1}^{\infty} \frac{a^{n}}{(1+a)(1+a^{2})\cdots(1+a^{n})} \quad (a>0)$
    • $\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1} \left( 1 - \sqrt[n]{a} \right) \quad (a > 0)$.

  • Estimate the error when $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^4}$ is approximated by the first $10$ terms.

  • What is the error if we approximate a convergent geometric series $\displaystyle \sum_{n=0}^{\infty} ar^{n}$ by the first $n$ terms $\displaystyle a + ar + \cdots + ar^{n-1}$?

  • If the series $\displaystyle \sum_{n=1}^{\infty} (a_n - a_{n-1})$ converges, and $\displaystyle \sum_{n=1}^{\infty} b_n$ converges absolutely. Prove that the series $\displaystyle \sum_{n=1}^{\infty} a_n b_n$ is absolutely convergent.

  • Suppose the positive-term series $\displaystyle \sum_{n=1}^{\infty} a_n$ is convergent. Prove that there exists a convergent positive-term series $\displaystyle \sum_{n=1}^{\infty} b_n$ such that $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = 0$.

  • If  $\displaystyle \sum_{n=1}^{\infty} a_n^{2}$ converges and $b, c$ are positive constants, does $\displaystyle \sum_{n=1}^{\infty} \frac{a_n^{2}}{1 + a_n^{2}}$ converge?

  • Let $f_n(x) = x^n \ln x$, find $\displaystyle \lim_{n \to \infty} \frac{1}{n!} f_n^{(n-1)}\left(\frac{1}{n}\right)$.

  • Given any series $\displaystyle \sum a_n$, we define a series $\displaystyle \sum a_n^{+}$ whose terms are all the positive terms of $\displaystyle \sum a_n$, and a series $\displaystyle \sum a_n^{-}$ whose terms are all the negative terms of $\displaystyle \sum a_n$. To be specific, we let \(a_n^{+} = \frac{a_n + \lvert a_n\rvert}{2}, \quad a_n^{-} = \frac{a_n - \lvert a_n\rvert}{2}.\)
    1. If $\displaystyle \sum a_n$ is absolutely convergent, show that both of the series $\displaystyle \sum a_n^{+}$ and $\displaystyle \sum a_n^{-}$ are convergent.
    2. If $\displaystyle \sum a_n$ is conditionally convergent, show that both of the series $\displaystyle \sum a_n^{+}$ and $\displaystyle \sum a_n^{-}$ are divergent.

  • Prove that if $\displaystyle \sum a_n$ is a conditionally convergent series and $r$ is any real number, then there is a rearrangement of $\displaystyle \sum a_n$ whose sum is $r$. (Hints: Take just enough positive terms $a_n^{+}$ so that their sum is greater than $r$. Then add just enough negative terms $a_n^{-}$ so that the cumulative sum is less than $r$. Continue in this manner and use the Sandwich Theorem.)

  • Discuss the value of $\lambda$ for which the following infinite series converges $\displaystyle\sum_{n=1}^{\infty} \frac{\ln\left(1 + \frac{1}{n^{\lambda}}\right)}{n}$.

  • Let $a_n$ ($n = 1, 2, \dots$) be the area of the planar region bounded by the curves $\displaystyle y = \frac{1}{x^{3}}$ ($x > 0$), $y = \dfrac{x}{n^{4}}$, and $\displaystyle y = \frac{x}{(n+1)^{4}}$. Find the sum of the series $\displaystyle \sum_{n=1}^{\infty} a_n$.