Sequences and Series – Convergence Tests, Power Series, Taylor Series & Fourier Series | Engineering Mathematics Notes for University & GATE

ENGINEERING MATHEMATICS NOTES – SEQUENCES AND SERIES (Gate and University Exam Focused)

(Convergence • Tests • Power Series • Taylor Series • Fourier Series)

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1. SEQUENCES

1.1 Definition

A sequence is a function whose domain is the set of positive integers.

$$a_1,a_2,a_3,\dots ,a_n,\dots$$

Examples:

$$a_n=\frac{1}{n},a_n=(-1{)}^n,a_n=\frac{n+1}{2n}$$

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1.2 Convergence of a Sequence

A sequence 

$\{a_n\}$ converges to $L$ if:

$$\underset{n\to \mathrm{\infty }}{\lim }{a}_n=L$$

If such L does not exist → divergent.

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1.3 Standard Convergent Sequences

• $\frac{1}{n} \to 0$
  

• $\frac{1}{n^p} \to 0$, $p>0$

• $(1+\frac1n)^n \to e$
  

• $\sqrt[n]{n} \to 1$
  

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1.4 Divergent Sequences

• $n, n^2, \sqrt{n}$
  

• $(-1)^n$ oscillates → divergent

• $\sin n$ has no limit → divergent

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1.5 Monotone Convergence Theorem

If a sequence is:

• monotonic increasing, and

• bounded above

→ It must converge.

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2. SERIES

A series is the sum of a sequence:

$$\sum _{n=1}^{\mathrm{\infty }}{a}_n$$

The partial sum:

$$S_N=\sum _{n=1}^N{a}_n$$

Series converges if:

$$\underset{N\to \mathrm{\infty }}{\lim }{S}_N=S$$

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3. CONVERGENCE OF SERIES (GENERAL RESULTS)

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3.1 Necessary Condition

If series converges:

$$\underset{n\to \mathrm{\infty }}{\lim }{a}_n=0$$

If this limit ≠ 0 → series diverges immediately.

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3.2 Geometric Series

$$\sum a{r}^n=\{\begin{array}{ll}\frac{a}{1-r}& \mathrm{\mid }r\mathrm{\mid }<1\\ \text{diverges}& \mathrm{\mid }r\mathrm{\mid }\ge 1\end{array}$$

Examples:

• $1+\frac12+\frac14+\cdots = 2$
  

• $2+1+\frac12+\cdots = 4$
  

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3.3 Harmonic Series

$$\sum \frac{1}{n} = \infty$$

divergent, even though terms → 0.

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3.4 p-Series

$$\sum \frac{1}{n^{\mathrm{p}}}$$

• convergent if p > 1

• divergent if $p \le 1$
  

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4. SERIES WITH NON-NEGATIVE TERMS

(GATE IMPORTANT SECTION)

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4.1 Comparison Test

Let $0 \le a_n \le b_n$

• If $\sum b_n$ converges → $\sum a_n$ converges

• If $\sum a_n$ diverges → $\sum b_n$ diverges

4.2 Limit Comparison Test

$$L = \lim_{n\to\infty} \frac{a_n}{b_n}$$

• If $0<L<\infty$, both converge or diverge together.

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4.3 Ratio Test

$$L = \lim_{n\to\infty} \frac{a_{n+1}}{a_n}$$

L	Result
(L < 1)	Convergent
(L > 1)	Divergent
(L = 1)	Inconclusive

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4.4 Root Test

$$L=\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{a_n}$$

Same decision table as ratio test.

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4.5 Integral Test

If:

• $a_n=f(n),$

• $f(x)$ is positive, decreasing, continuous on $[1,\infty)$
  

Then:

$$\sum_1^\infty a_n \text{ converges } \iff \int_1^\infty f(x)\,dx \text{ converges}$$

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Example (Integral Test)

Test convergence of:

$$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n(\ln n{)}^{\mathrm{2}}}$$

Compare integral:

$$\int \frac{dx}{x(\ln x{)}^2}$$

Let $u = \ln x$:

$$\int \frac{1}{u^2}du$$

Convergent → series convergent.

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5. POWER SERIES

5.1 Definition

A power series centered at a:

$$\sum _{n=0}^{\mathrm{\infty }}{c}_n(x-a{)}^n$$

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5.2 Radius of Convergence (R)

Use ratio test:

$$R=\frac{1}{\lim \mathrm{\mid }{c}_{n+1}\mathrm{/}{c}_n\mathrm{\mid }}$$

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5.3 Interval of Convergence

$$(a-R,a+R)$$

Test endpoints separately.

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5.4 Standard Power Series

$$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$ $$\sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$ $$\cos x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}$$ $$\frac{1}{1-x}=\sum x^n,\mathrm{\mid }x\mathrm{\mid }<1$$

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6. TAYLOR SERIES

6.1 Taylor Series Expansion

$$f(x)=f(a)+f^{\mathrm{\prime }}(a)(x-a)+\frac{f^{\mathrm{\prime }\mathrm{\prime }}(a)}{2!}(x-a{)}^2+\cdots$$

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Examples

1. Expand 

$e^x$ at $x=0$

$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$$

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6.2 Maclaurin Series (a = 0)

Special case of Taylor series.

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6.3 Error Term (Lagrange Form)

$$R_n(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-a{)}^{n+1}$$

Used for truncation error estimation.

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7. FOURIER SERIES

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7.1 Periodic Function

Function $f(x)$ with period $2\pi$ satisfies:

$$f(x+2\pi )=f(x)$$

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7.2 Fourier Series of f(x)

$$f(x) = a_0 + \sum_{n=1}^\infty (a_n \cos nx + b_n \sin nx)$$

Where:

$$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)\, dx$$
$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx)\, dx$$
$$b_n=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin (nx)dx$$

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7.3 EVEN & ODD FUNCTIONS

Important shortcuts:

If $f(x)$ is even:

• All $b_n = 0$
  

• Only cosine terms appear.

If $f(x)$ is odd:

• All $a_n = 0$
  

• Only sine terms appear.

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7.4 Standard Fourier Series Examples

Square Wave (Odd Function)

$$f(x)= \begin{cases} 1 & 0<x<\pi\\ -1 & -\pi < x < 0 \end{cases}$$ $$f(x)=\frac{4}{\pi }(\sin x+\frac{\sin 3x}{3}+\frac{\sin 5x}{5}+\cdots )$$

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Sawtooth Wave

$$f(x)=x,\quad -\pi<x<\pi$$
$$f(x)=-2\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n}\sin (nx)$$

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8. GATE EXAM SHORTCUTS

✔ Ratio and Root test are fastest for exponential-type terms

✔ For p-series always check p > 1

✔ Power series radius = ratio test on coefficients

✔ Fourier series: always check even/odd first

✔ Taylor series: memorize expansions of 

$e^x, \sin x, \cos x, (1+x)^n$

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9. COMMON MISTAKES

❌ Assuming convergence because terms → 0
❌ Ignoring endpoint tests in power series
❌ Forgetting that integral test requires positive decreasing functions
❌ Applying ratio test to alternating series (use alternating test instead)

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10. LAST-MINUTE REVISION NOTES

• Sequence converges if limit exists

• Series converges if partial sums converge

• p-series convergent if p>1

• Ratio test: L<1 converge

• Root test: same rule

• Power series: radius from ratio test

• Taylor expansion: derivatives at point

• Fourier series: compute (a_0, a_n, b_n)

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