Dr. Xiaozhou Li

Dr. Xiaozhou Li

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Lecture Note: §11 Infinite Sequences and Series

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Last Update: 2026-03-22

§10.1 Sequences

Sequences are fundamental to the study of infinite series and to many aspects of mathematics.

  • Infinite sequences $a_1, a_2, a_3, \ldots, a_n, \ldots$
    • Explicit definition: writing rules that specify the $n$th terms $a_n = f(n)$.
    • Recursive definition.
  • Limit of sequence $\displaystyle \lim_{n \to \infty} a_n$, similar as $\displaystyle \lim_{x \to \infty} f(x)$.
    • The sequence ${a_n}$ converges to the number $L$ if for every positive number $\varepsilon$ there corresponds an integer $N$ such that \(|a_n - L| < \varepsilon \quad \text{whenever} \quad n > N.\) If no such number $L$ exists, we say that ${a_n}$ diverges.
    • If ${a_n}$ converges to $L$, we write \(\lim_{n \to \infty} a_n = L.\)
  • Properties of Convergent Sequences
    • Limit laws: sum, difference, constant multiple, product, quotient rules.
    • Squeeze theorem (sandwich theorem)
    • Continuous function theorem: $\displaystyle \lim_{n \to \infty} f(a_n) = f(\lim_{n\to \infty}a_n)$.
  • Monotonic and Bounded Sequences
    • Bounded Sequences
      • Completeness theorem of real number: bounded $\Rightarrow$ least upper bounded and greatest lower bound. The fact that there is no gap or hole in the real number
    • Monotonic Sequences
    • Monotonic sequence theorem
      • bounded from above $+$ nondecreasing $\Rightarrow$ convergence
      • bounded from below $+$ nonincreasing $\Rightarrow$ convergence

§10.2 Infinite Series

  • Infinite series $\displaystyle \sum_{n =0}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots+ a_n+ \cdots$

  • Convergence of series
    • Partial sum sequence ${s_n}$, where $s_n = a_1 + a_2 + \cdots + a_n$.
    • $\displaystyle \sum_{n=1}^{\infty} a_n= \lim_{n \to \infty} s_n = \lim\limits_{n \to \infty} \sum_{i=1}^n a_i$.
    • From the perspective of convergence, infinite series and infinite sequences are essentially the same.
  • Classic Series (the formula $s_n$ can be easily found)
    • Geometric series: $\displaystyle \sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1 - r}, \quad \lvert r\rvert < 1$.
    • “Telescoping” series: most of the terms cancel in each of the partial sums.
  • Properties for convergent series
    • Operation rules: sum, difference, constant multiple rules.
      • If $\sum a_n$ and $\sum b_n$ converge, then $\sum (a_n + b_n)$ and $\sum (a_n - b_n)$ both converge.
      • If $\sum a_n$ converges and $\sum b_n$ diverges, then $\sum (a_n + b_n)$ and $\sum (a_n - b_n)$ both diverge.
      • Every nonzero constant multiple of a convergent (divergent) series converges (diverges).
    • A finite number of terms doesn’t affect the convergence or divergence of a series.

  • A simple formula for the $s_n$ is not easy to discover in general $\Longrightarrow$ need to develop several tests that enable us to determine whether a series is convergent or divergent without explicitly finding its sum.

  • The $n$th term test for divergence
    • If $\displaystyle \sum_{n=1}^{\infty} a_n$ converges, then $a_n \to 0$.
    • $\displaystyle \sum_{n=1}^{\infty} a_n$ diverges if $\displaystyle \lim_{n \to \infty} a_n$ fails to exist or is different from zero.
    • converse of the theorem is not true in general: harmonic series

§10.3 The Integral Test

  • A series $\displaystyle \sum_{n=1}^{\infty} a_n$ of nonnegative terms ($a_n \geq 0$).
    • converges $\Longleftrightarrow$ its partial sums are bounded from above.

  • The relation between infinite series and improper integral.

  • The Integral Test
    • Series $\displaystyle \sum_{n=1}^{\infty} a_n$, $a_n = f(n) \geq 0$, where
    • $f$: a continuous, positive, decreasing function
    • Then the series $\displaystyle \sum_{n=N}^{\infty} a_n$ and the integral $\displaystyle \int_{N}^{\infty} f(x)dx$ both converge or both diverge.
  • Classic series: $p$-series
    • $\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^{p}} =1 + \frac{1}{2^{p}} + \frac{1}{3^{p}} + \frac{1}{4^{p}} + \cdots + \frac{1}{n^{p}} + \cdots$
    • converges if $p > 1$ and diverges if $p \leq 1$.
  • Estimating the sum of a series
    • how good is the partial sum $s_n$ as an approximation for the real sum $S$.
    • Remainder $R_n = S - s_n$ estimate for the integral test: $\displaystyle \int_{n+1}^{\infty} f(x)\,dx \le R_n \le \int_{n}^{\infty} f(x)\,dx.$

§10.4 Comparison Tests

  • The idea: compare a given series with a series that is known to be convergent or divergent, such as
    • geometric series
    • $p$-series
  • The Direct Comparison Test, $\sum a_n$ and $\sum b_n$ are series with positive terms
    • If $\sum b_n$ is convergent and $a_n \leq b_n$, then $\sum a_n$ is also convergent
    • If $\sum b_n$ is divergent and $a_n \geq b_n$, then $\sum a_n$ is also divergent
  • The Limit Comparison Test, $\sum a_n$ and $\sum b_n$ are series with positive terms
    • If $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = c > 0$, then either both series converge or both diverge
    • If $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = 0$ and $\sum b_n$ converges, then $\sum a_n$ converges.
    • If $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = \infty$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

§10.5 Absolute Convergence, The Ratio and Root Tests

  • Absolute convergence:
    • Some of the terms of a series are positive and others are negative. However, $\sum \lvert a_n\rvert$ is a series of positive terms.
    • A series converges absolutely (is absolutely convergent) if the series of absolute values $\sum \lvert a_n \rvert$ is convergent.

  • The Absolute Convergence Test - If $\sum \lvert a_n \rvert$ converges, then $\sum a_n$ converges.

  • The Ratio Test. Let $\sum a_n$ be any series and suppose that \(\lim_{n \to \infty} \left\lvert \frac{a_{n+1}}{a_n} \right\rvert = \rho.\) Then
    1. the series converges absolutely if $\rho < 1$,
    2. the series diverges if $\rho > 1$ or $\rho$ is infinite,
    3. the test is inconclusive if $\rho = 1$.
  • The Root Test. Let $\sum a_n$ be any series** and suppose that \(\lim_{n \to \infty} \sqrt[n]{\lvert a_n \rvert} = \rho.\) Then
    1. the series converges absolutely if $\rho < 1$,
    2. the series diverges if $\rho > 1$ or $\rho$ is infinite,
    3. the test is inconclusive if $\rho = 1$.

§10.6 Alternating Series, Conditional Convergence

  • Alternating Series
    • $a_n = (-1)^{n+1}\lvert a_n \rvert$ or $a_n = (-1)^{n} \lvert a_n \rvert$
  • The Alternating Series Test
    • The alternating series converges if $\lvert a_n\rvert$ is nonincreasing can converges to $0$.
  • Conditional Convergence
    • A series that is convergent but not absolutely convergent is called conditionally convergent.

§ Strategy for Testing Series

We now have several ways of testing a series for convergence or divergence; the problem is to decide which test to use on which series. The main strategy is to classify the series according to its form.

  1. Is the series geometric?
  2. Is the series $p$-series?
  3. Do the terms go to $0$? Try the $n$th term test.
  4. Are there negative terms in the series? If so, you may have to use the absolute convergence test or the alternating series test.
  5. Are factorials involved? If so, use the ratio test. The test is also useful when there are exponentials involved but the series isn’t geometric.
  6. Are there tricky exponentials with $n$ in the base and the exponent? If so, try the root test. In general, if it is easy to take the $n$th root of the term $a_n$, the root test is probably a winner.
  7. Do the terms have a factor of exactly $1/n$ as well as logarithms? In that case, the integral test is probably what you want.
  8. Do none of the above tests seem to work? You may have to use the comparison test or the limit comparison test in conjunction with the $p$-series, as well as all the understanding of the behavior of functions

§10.7 Power Series

  • A power series at $x = a$: \(\sum_{n=0}^{\infty} c_n (x - a)^{n} = c_0 + c_1 (x - a) + c_2 (x - a)^{2} + \cdots + c_n (x - a)^{n} + \cdots\)
    • Where power series converge or diverge?
      • Radius of convergence.
      • The interval of convergence
  • The Convergence Theorem for Power Series. The convergence of the power series $\sum c_n(x-a)^{n}$ is described by one of the following three cases:
    1. There is a positive number $R$ such that the series diverges for $x$ with $\lvert x-a\rvert > R$ but converges absolutely for $x$ with $\lvert x-a\rvert < R$. The series may or may not converge at either of the endpoints $x = a-R$ and $x = a+R$.
    2. The series converges absolutely for every $x$ ($R = \infty$).
    3. The series converges at $x = a$ and diverges elsewhere ($R = 0$).
  • How to Test a Power Series for Convergence
    1. Use the Ratio Test or the Root Test to find the largest open interval where the series converges absolutely, \(\lvert x-a\rvert < R \quad \text{or} \quad a-R < x < a+R.\)
    2. If $R$ is finite, test for convergence or divergence at each endpoint using other tests.
    3. If $R$ is finite, the series diverges for $\lvert x-a\rvert > R$.
  • Operations of Power Series, for $\lvert x \rvert <R$.
    • Series multiplication
    • Series division
    • Function substitution
    • Term-by-term differentiation
    • Term-by-term integration
  • Classic problem: find the sum of the power series.

§10.8 Taylor and Maclaurin Series

Important Question: can we represent function $f(x)$ as the sum of power series?

  • Taylor Series generated by $f$ at $x = a$, \(\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k !}(x-a)^k = f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2} +\cdots+\frac{f^{(n)}(a)}{n !}(x-a)^{n}+\cdots\)

    • No guarantee that the sum of the Taylor series is equal to $f$.
    • The power series representation at $a$ of a function is unique, regardless of how it is found.

  • Taylor polynomial of order $n$, \(P_n(x)\sum_{k=0}^{n} \frac{f^{(k)}(a)}{k !}(x-a)^{k}\)

  • Classic problem: find the Taylor series of $f(x)$.

    • $\displaystyle e^{x} = 1 + x + \cdots + \frac{x^{n}}{n !} + \cdots$
    • $\displaystyle \sin x = x - \frac{x^{3}}{3 !} + \cdots + (-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!} + \cdots$
    • $\displaystyle \cos x = 1 - \frac{x^{2}}{2!} + \cdots + (-1)^n\frac{x^{2n}}{(2n)!} + \cdots$

§10.9 Convergence of Taylor Series

When is a function represented by its Taylor series?

  • The remainder (error term) of the Taylor series:
    • $\displaystyle R_n(x) = f(x) - P_n(x)$
  • Taylor’s Theorem (Taylor’s Formula):
    • $\displaystyle R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x - a)^{n+1}$
    • If $R_n(x) \rightarrow 0$ as $n \rightarrow \infty$ for all $x \in I$, \(f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k !}(x-a)^{k}\).
  • Estimating the Remainder
    • If $\lvert f^{(n+1)}(x)\rvert \leq M$, then the reminder term satisfies the inequality $\displaystyle R_n(x) \leq M\frac{\lvert x-a \rvert^{n+1}}{(n+1)!}$.
  • New Taylor series from old.

§10.10 Applications of Taylor Series

  • The Binomial Series: for $-1 < x < 1$, \(\displaystyle (1+x)^{m}=1+ mx + \frac{m(m-1)}{2!}x^{2} + \cdots + \frac{m(m-1)\cdots(m-k+1)}{k!} x^{k} +\cdots\)

  • Leibniz’s formula: $\displaystyle \frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9} \cdots+\frac{(-1)^n}{2 n+1}+\cdots$

  • Some applications of Taylor series
    • Evaluating Non-elementary Integrals, such as $\displaystyle \int \sin x^{2} \, dx$,$\displaystyle \int e^{-x^{2}}\, dx$
    • Approximate the definite integral
    • Evaluating Indeterminate Forms
    • Find the $n$th derivatives
    • Find the sum of series
    • Find the solution of differentition equation
  • Euler’s Identity: for any real number $\theta$, $\displaystyle e^{i\theta} = \cos\theta + i \sin\theta$.

Frequently Used Taylor Series

  • $\displaystyle \frac{1}{1 - x} = 1 + x + x^{2} + \cdots + x^{n} + \cdots = \sum_{n=0}^{\infty} x^{n}, \quad \lvert x\rvert < 1$
  • $\displaystyle \frac{1}{1 + x} = 1 - x + x^{2} - \cdots + (-1)^{n}x^{n} + \cdots = \sum_{n=0}^{\infty} (-1)^{n}x^{n}, \quad \lvert x\rvert < 1$
  • $\displaystyle e^{x} = 1 + x + \frac{x^{2}}{2!} + \cdots + \frac{x^{n}}{n!} + \cdots = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}, \quad \lvert x\rvert < \infty$
  • $\displaystyle \sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots + (-1)^{n} \frac{x^{2n+1}}{(2n+1)!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n+1}}{(2n+1)!}, \quad \lvert x\rvert < \infty$
  • $\displaystyle \cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots + (-1)^{n} \frac{x^{2n}}{(2n)!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!}, \quad \lvert x\rvert < \infty$
  • $\displaystyle \ln(1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \cdots + (-1)^{n-1} \frac{x^{n}}{n} + \cdots = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^{n}}{n}, \quad -1 < x \leq 1$
  • $\displaystyle \arctan x = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \cdots + (-1)^{n} \frac{x^{2n+1}}{2n+1} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n+1}}{2n+1}, \quad \lvert x\rvert \leq 1$
  • $\displaystyle (1+x)^{m} = 1 + mx + \frac{m(m-1)}{2}x^{3} + \cdots + {m\choose n}x^{n} + \cdots,\quad \lvert x\rvert < 1$