Sequences & Series: Convergence, Tests, Power Series & Applications

1. Sequences: Basics and Convergence

1.1 Definition of a Sequence

A sequence is a function $a_n=f(n)$ whose domain is the set of natural numbers $\mathbb{N}$. It is written as: $$a_1,a_2,a_3,\dots ,a_n,\dots$$

• Example: $a_n=\frac{1}{n}$ → $1,\frac{1}{2},\frac{1}{3},\dots$

1.2 Convergence of a Sequence

A sequence $\{a_n\}$ converges to a limit $L$ if: $$\underset{n\to \infty }{\lim }{a}_n=L$$ This means for every $ϵ>0$, there exists $N\in \mathbb{N}$ such that for all $n\ge N$, $|a_n-L|<ϵ$.

• Divergence: If no such $L$ exists, the sequence diverges (e.g., $a_n=n$ diverges to $+\infty$).

Worked Example: Proving Convergence

Problem: Show $a_n=\frac{3n^2+2}{5n^2-1}$ converges and find its limit. Solution: Divide numerator and denominator by $n^2$: $$a_n=\frac{3+\frac{2}{n^2}}{5-\frac{1}{n^2}}\to \frac{3+0}{5-0}=\frac{3}{5}\text{as}n\to \infty .$$ Conclusion: ${\lim }_{n\to \infty }{a}_n=\frac{3}{5}$.

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2. Series: Definition and Basic Tests

2.1 Definition of a Series

A series is the sum of the terms of a sequence: $$\sum _{n=1}^{\infty }{a}_n=a_1+a_2+a_3+\dots$$

• If $S_N={\sum }_{n=1}^N{a}_n$ converges to $S$ as $N\to \infty$, the series converges to $S$.
• If $S_N$ diverges, the series diverges.

2.2 Divergence Tests

1. n-th Term Test (Divergence Test):

   • If ${\lim }_{n\to \infty }{a}_n\ne 0$, the series diverges.
   • Note: If ${\lim }_{n\to \infty }{a}_n=0$, the test is inconclusive (series may converge or diverge).

   Example: ${\sum }_{n=1}^{\infty }\frac{n}{n+1}$ diverges because ${\lim }_{n\to \infty }\frac{n}{n+1}=1\ne 0$.

2. Comparison Test:

   • If $0\le a_n\le b_n$ for all $n$ and $\sum b_n$ converges, then $\sum a_n$ converges.
   • If $a_n\ge b_n\ge 0$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

   Example: Compare $\sum \frac{1}{n^2+1}$ with $\sum \frac{1}{n^2}$ (known to converge). Since $\frac{1}{n^2+1}\le \frac{1}{n^2}$, the series converges.

2.3 Ratio and Root Tests

Example (Ratio Test): Test ${\sum }_{n=1}^{\infty }\frac{n!}{10^n}$. $$L=\underset{n\to \infty }{\lim }\frac{(n+1)!/10^{n+1}}{n!/10^n}=\underset{n\to \infty }{\lim }\frac{n+1}{10}=\infty >1⟹\text{diverges}.$$

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3. Taylor and Maclaurin Series

3.1 Taylor Series

The Taylor series of $f(x)$ centered at $x=a$ is: $$f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n,$$ where $f^{(n)}(a)$ is the $n$-th derivative of $f$ at $x=a$.

3.2 Maclaurin Series (Special Case)

When $a=0$, the series becomes the Maclaurin series: $$f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(0)}{n!}{x}^n.$$

Common Maclaurin Series

Worked Example: Maclaurin Series for $e^x$

Problem: Find the Maclaurin series for $e^x$ and prove it converges to $e^x$ for all $x$. Solution:

1. Compute derivatives at $x=0$: $f(x)=e^x⟹f(0)=1$, $f^{\prime }(x)=e^x⟹f^{\prime }(0)=1$, $f^{\prime \prime }(x)=e^x⟹f^{\prime \prime }(0)=1$, and so on.
2. The Maclaurin series is: $$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}.$$
3. Proof of Convergence: The remainder term $R_n(x)$ of the Taylor series satisfies: $$R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}{x}^{n+1}=\frac{e^c}{(n+1)!}{x}^{n+1},$$ where $c$ is between $0$ and $x$. Since $e^c\le e^{|x|}$, and $\frac{|x{|}^{n+1}}{(n+1)!}\to 0$ as $n\to \infty$, the series converges to $e^x$ for all $x$.

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4. Power Series and Convergence

4.1 Definition

A power series is of the form: $$\sum _{n=0}^{\infty }{c}_n(x-a{)}^n,$$ where $c_n$ are coefficients and $a$ is the center.

4.2 Radius and Interval of Convergence

• Radius of Convergence ($R$): The distance from $a$ within which the series converges.
  • Found using the Ratio Test: $$R=\underset{n\to \infty }{\lim }\left|\frac{c_n}{c_{n+1}}\right|.$$
• Interval of Convergence: $(a-R,a+R)$. Test endpoints separately.

Worked Example: Radius of Convergence

Problem: Find the radius of convergence for ${\sum }_{n=1}^{\infty }\frac{(x-3{)}^n}{n}$. Solution: Apply the Ratio Test: $$L=\underset{n\to \infty }{\lim }\left|\frac{(x-3{)}^{n+1}/(n+1)}{(x-3{)}^n/n}\right|=\underset{n\to \infty }{\lim }\left|\frac{n}{n+1}(x-3)\right|=|x-3|.$$ For convergence, $L<1⟹|x-3|<1⟹R=1$. Interval: $(2,4)$. Check endpoints:

• At $x=2$: $\sum \frac{(-1{)}^n}{n}$ (converges by Alternating Series Test).
• At $x=4$: $\sum \frac{1}{n}$ (diverges by Harmonic Series Test). Final Interval: $[2,4)$.

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5. Applications of Series

5.1 Approximating Functions

Use Taylor/Maclaurin series to approximate functions near a point. Example: Approximate $\sin (0.1)$ using the first 3 terms of its Maclaurin series. $$\sin x\approx x-\frac{x^3}{6}⟹\sin (0.1)\approx 0.1-\frac{0.001}{6}\approx \mathrm{0.09983.}$$ (The actual value is $\approx 0.099833$.)

5.2 Solving Differential Equations

Power series are used to find solutions to differential equations (e.g., Bessel’s equation). Example: Assume $y(x)={\sum }_{n=0}^{\infty }{a}_n{x}^n$ and substitute into $y^{\prime \prime }+xy=0$.

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6. Exam Tip: Key Strategies for TU/PU Exams

1. Convergence Tests:

   • Always check the n-th term test first (quick elimination).
   • For positive-term series, use Comparison Test or Ratio Test.
   • For alternating series, use the Alternating Series Test ($|a_{n+1}|\le |a_n|$ and $\lim a_n=0$).

2. Taylor/Maclaurin Series:

   • Memorize the standard expansions for $e^x,\sin x,\cos x,\frac{1}{1-x}$.
   • For proving convergence, use the remainder term $R_n(x)$.

3. Power Series:

   • Find the radius of convergence using the Ratio Test.
   • Always test endpoints separately (they may converge or diverge).

4. Common Mistakes to Avoid:

   • Forgetting to check endpoints in power series.
   • Misapplying the Ratio Test (e.g., not taking the limit correctly).
   • Assuming a series converges just because $\lim a_n=0$ (n-th term test is inconclusive in this case).

5. Past Exam Patterns:

   • 20% weightage: Proving convergence/divergence using tests.
   • 30% weightage: Finding Taylor/Maclaurin series and approximations.
   • 25% weightage: Power series and radius of convergence.
   • 25% weightage: Applications (e.g., approximating functions).

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Summary Table: Convergence Tests

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Practice Problems (TU-Style)

1. Determine convergence: ${\sum }_{n=1}^{\infty }\frac{n^2}{5n^2+4}$. Hint: Compare with $\sum \frac{1}{5}$.

2. Find Maclaurin series for $\cos x$ and prove its convergence for all $x$.

3. Find the radius of convergence for ${\sum }_{n=1}^{\infty }\frac{(x+2{)}^n}{n^2}$.

4. Approximate $\ln (1.1)$ using the Taylor series of $\ln (1+x)$ centered at $x=0$ (first 3 terms).

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References for Further Study

• Textbook: Thomas’ Calculus (Chapter 11: Infinite Series).
• Online: Khan Academy (Series and Convergence Tests).
• TU Past Papers: Focus on 2018–2023 questions for pattern recognition.