Mathematics I (BT-102) Unit 3 Notes - RGPV

How unit 3 is examined

Convergence of sequences and series, tests, power and Taylor series, standard expansions, and Fourier half range series with Parseval's theorem; the paper asks Fourier series (x+x^2, e^x, x(pi-x)), a geometric-series convergence and a Taylor expansion of sin x.

Convergence of sequence and series

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Definition. A sequence $\{a_n\}$ converges if $\lim_{n\to\infty}a_n=l$ is finite; a series $\sum a_n$ converges if its partial sums $S_n=a_1+\dots+a_n$ tend to a finite limit, otherwise it diverges or oscillates.

Key points.

A convergent series must have $a_n\to0$, but $a_n\to0$ alone does not guarantee convergence (for example $\sum 1/n$ diverges).
The geometric series $a+ar+ar^2+\dots$ converges to $\dfrac{a}{1-r}$ if $|r|<1$ and diverges if $|r|\ge1$.
The p-series $\sum 1/n^p$ converges if $p>1$ and diverges if $p\le1$.

Example. $\frac14+\frac1{4^2}+\frac1{4^3}+\dots$ is geometric with $a=r=\frac14<1$, so it converges to $\dfrac{1/4}{1-1/4}=\dfrac13$.

A sequence that is monotonic and bounded converges; a series of positive terms converges exactly when its partial sums are bounded above.
Adding or removing finitely many terms does not change convergence, and $\sum a_n$ converging absolutely ($\sum|a_n|$ converges) implies it converges.
To show the geometric series converges, write $S_n=\dfrac{a(1-r^n)}{1-r}$ and note $r^n\to0$ when $|r|<1$, so $S_n\to\dfrac{a}{1-r}$.

Asked: [7 marks] (Dec 2024) Show that the series $\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+\dots+\frac{1}{4^n}+\dots$ is convergent.

Tests for convergence

Definition. A test decides convergence of $\sum u_n$ (positive terms) without finding the sum.

Key points.

Comparison test: if $0\le u_n\le v_n$ and $\sum v_n$ converges then $\sum u_n$ converges; if $\sum u_n$ diverges then $\sum v_n$ diverges.
Ratio test: if $\lim u_{n+1}/u_n=l$, the series converges for $l<1$, diverges for $l>1$, and the test fails for $l=1$.
Root test: if $\lim (u_n)^{1/n}=l$, the same conditions on $l$ apply.
Leibnitz test: an alternating series $\sum(-1)^{n}u_n$ converges if $u_n$ decreases and $u_n\to0$.
Harmonic series $\sum1/n$ diverges, while $\sum1/n^2$ converges; use them as comparison series.
Limit comparison: if $\lim u_n/v_n=k$ is finite and non-zero, then $\sum u_n$ and $\sum v_n$ converge or diverge together.
Cauchy's integral test: $\sum f(n)$ and $\int_1^\infty f(x)dx$ converge or diverge together for positive decreasing $f$.
Raabe's test is used when the ratio test gives $l=1$: with $\lim n\left(\dfrac{u_n}{u_{n+1}}-1\right)=l$, the series converges if $l>1$ and diverges if $l<1$.

Example. For $\sum\dfrac{n}{2^n}$, $\dfrac{u_{n+1}}{u_n}=\dfrac{n+1}{2n}\to\dfrac12<1$, so it converges by the ratio test.

Power series

Definition. A power series is $\sum_{n=0}^\infty a_n(x-c)^n$, a series in powers of $x-c$.

Key points.

It always converges at $x=c$, and it has a radius of convergence $R=\lim\left|\dfrac{a_n}{a_{n+1}}\right|$.
It converges absolutely for $|x-c|<R$ and diverges for $|x-c|>R$; the end points must be tested separately.
Inside the interval it may be differentiated and integrated term by term.
Example: $\sum\dfrac{x^n}{n!}$ has $a_n/a_{n+1}=n+1\to\infty$, so $R=\infty$ and it converges for every $x$.
Example: $\sum x^n$ has $R=1$, converging for $-1<x<1$ and diverging at both end points.

Taylor's series

Definition. If $f$ has derivatives of all orders at $a$, then $$f(x)=f(a)+(x-a)f'(a)+\frac{(x-a)^2}{2!}f''(a)+\frac{(x-a)^3}{3!}f'''(a)+\dots$$ Putting $a=0$ gives Maclaurin's series.

Key points.

The series represents $f$ only where the remainder term tends to zero.
Find the derivatives at the point $a$, substitute in the formula and write the pattern of terms.
Writing $x=a+h$ gives the equivalent form $f(a+h)=f(a)+hf'(a)+\dfrac{h^2}{2!}f''(a)+\dots$
Lagrange's remainder after $n$ terms is $R_n=\dfrac{h^n}{n!}f^{(n)}(a+\theta h)$ with $0<\theta<1$.

Example. For $f=\sin x$ at $a=\pi/2$: $f=1,\ f'=\cos=0,\ f''=-1,\ f'''=0,\ f^{iv}=1$, so $$\sin x=1-\frac{(x-\pi/2)^2}{2!}+\frac{(x-\pi/2)^4}{4!}-\dots$$

Asked: [7 marks] (Dec 2024) Find the Taylor's expansion of $y=\sin x$ about the point $x=\pi/2$.

Series for exponential, trigonometric and logarithm functions

Definition. These are the Maclaurin series of the standard functions.

Key points.

$e^x=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dots$ for all $x$.
$\sin x=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots$ and $\cos x=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dots$ for all $x$.
$\log(1+x)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\dots$ valid for $-1<x\le1$.
$\log(1-x)=-\left(x+\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dots\right)$ for $-1\le x<1$.
Taylor's series is valid for the function where $R_n\to0$; for $\sin x$ and $\cos x$ that is every $x$.
Substituting $x\to-x$ or $x\to x^2$ in these series gives new expansions, for example $e^{-x}=1-x+\dfrac{x^2}{2!}-\dots$
$\tan^{-1}x=x-\dfrac{x^3}{3}+\dfrac{x^5}{5}-\dots$ for $-1\le x\le1$.
$\dfrac1{1-x}=1+x+x^2+\dots$ for $|x|<1$, and $\cosh x=1+\dfrac{x^2}{2!}+\dots$, $\sinh x=x+\dfrac{x^3}{3!}+\dots$

Example. $e^x\log(1+x)$: multiply $(1+x+\frac{x^2}2+\dots)(x-\frac{x^2}2+\frac{x^3}3)=x+\frac{x^2}{2}+\frac{x^3}{3}+\dots$

Fourier series: Half range sine and cosine series

Definition. A function $f(x)$ on $(-\pi,\pi)$ is expanded as $$f(x)=\frac{a_0}{2}+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)$$

Formula. $$a_0=\frac1\pi\int_{-\pi}^{\pi}f\,dx,\quad a_n=\frac1\pi\int_{-\pi}^{\pi}f\cos nx\,dx,\quad b_n=\frac1\pi\int_{-\pi}^{\pi}f\sin nx\,dx$$

Key points.

For an even function $b_n=0$ and the integrals run over $0$ to $\pi$ with a factor $2/\pi$; for an odd function $a_0=a_n=0$.
Half range sine series on $(0,\pi)$: $f=\sum b_n\sin nx$ with $b_n=\dfrac2\pi\int_0^\pi f\sin nx\,dx$.
Half range cosine series on $(0,\pi)$: $f=\dfrac{a_0}2+\sum a_n\cos nx$ with $a_n=\dfrac2\pi\int_0^\pi f\cos nx\,dx$.
On $(0,l)$ replace $nx$ by $n\pi x/l$ and $2/\pi$ by $2/l$.
At $x=\pi/2$ the sine series gives the deductions asked.

Steps.

Step 1: Note the interval and whether f is even, odd or neither; drop the zero coefficients.
Step 2: Find a0 = (1/pi) * integral of f dx, using the limits of the interval.
Step 3: Find a_n and b_n by integration by parts, using sin(n pi) = 0 and cos(n pi) = (-1)^n.
Step 4: Substitute in f = a0/2 + sum(a_n cos nx + b_n sin nx) and write the first few terms.

Integration by parts result used: $\int x\sin nx\,dx=-\dfrac{x\cos nx}{n}+\dfrac{\sin nx}{n^2}$ and $\int x^2\cos nx\,dx=\dfrac{x^2\sin nx}{n}+\dfrac{2x\cos nx}{n^2}-\dfrac{2\sin nx}{n^3}$.
Over $(-\pi,\pi)$ the term $x$ is odd and $x^2$ is even, so $x$ gives only $b_n$ and $x^2$ only $a_0,a_n$; this splits the work.
At a point of discontinuity the series converges to the mean $\frac12[f(x^-)+f(x^+)]$, and the half range sine series of $f$ is the Fourier series of its odd extension.

Example. For $f=x+x^2$ on $(-\pi,\pi)$: $a_0=\dfrac{2\pi^2}{3}$, $a_n=\dfrac{4(-1)^n}{n^2}$ (from $x^2$), $b_n=\dfrac{2(-1)^{n+1}}{n}$ (from $x$), so $$x+x^2=\frac{\pi^2}{3}+\sum_{n=1}^\infty\left[\frac{4(-1)^n}{n^2}\cos nx+\frac{2(-1)^{n+1}}{n}\sin nx\right]$$

Example. $f=x(\pi-x)$ on $(0,\pi)$: $b_n=\dfrac{2}{\pi}\cdot\dfrac{2(1-(-1)^n)}{n^3}=\dfrac{8}{\pi n^3}$ for odd $n$, zero for even $n$, so $x(\pi-x)=\dfrac8\pi\sum_{n\,odd}\dfrac{\sin nx}{n^3}$. Put $x=\pi/2$: $\dfrac{\pi^2}{4}=\dfrac8\pi\left(1-\dfrac1{3^3}+\dfrac1{5^3}-\dots\right)$, hence $$\frac1{1^3}-\frac1{3^3}+\frac1{5^3}-\dots=\frac{\pi^3}{32}$$

Example. $f=e^x$ on $(0,1)$, sine series with $l=1$: $b_n=2\int_0^1e^x\sin n\pi x\,dx=\dfrac{2n\pi\,(1-e(-1)^n)}{1+n^2\pi^2}$, so $e^x=\sum b_n\sin n\pi x$.

Working for $x(\pi-x)$. $b_n=\dfrac2\pi\int_0^\pi(\pi x-x^2)\sin nx\,dx=\dfrac2\pi\left[\dfrac{2}{n^3}-\dfrac{2\cos n\pi}{n^3}\right]=\dfrac{4(1-(-1)^n)}{\pi n^3}$, since the boundary terms vanish at $0$ and $\pi$.

Working for $e^x$. Use $\int e^{x}\sin n\pi x\,dx=\dfrac{e^{x}(\sin n\pi x-n\pi\cos n\pi x)}{1+n^2\pi^2}$ between $0$ and $1$, with $\sin n\pi=0$ and $\cos n\pi=(-1)^n$.

Example. $f=x$ on $(0,\pi)$: sine series $b_n=\dfrac{2(-1)^{n+1}}{n}$, cosine series $a_0=\pi$, $a_n=\dfrac{2((-1)^n-1)}{\pi n^2}$, that is $-\dfrac4{\pi n^2}$ for odd $n$.

Asked: [7 marks] (Jun 2025) Find the Fourier series for $f(x)=x+x^2$ in $(-\pi,\pi)$. Asked: [7 marks] (Dec 2024) Obtain the half range sine series for $f(x)=e^x$ in $0<x<1$. Asked: [7 marks] (Jun 2025) Find the half range sine series for $f(x)=x(\pi-x)$ in $(0,\pi)$; hence deduce $\frac1{1^3}-\frac1{3^3}+\frac1{5^3}-\dots=\frac{\pi^3}{32}$.

Parseval's theorem

Definition. ==The mean square value of $f$ equals the sum of the squares of its Fourier coefficients: $\dfrac1\pi\int_{-\pi}^{\pi}[f(x)]^2dx=\dfrac{a_0^2}{2}+\sum_{n=1}^\infty(a_n^2+b_n^2)$.==

Key points.

For a half range sine series on $(0,\pi)$: $\dfrac2\pi\int_0^\pi f^2dx=\sum b_n^2$.
For a half range cosine series on $(0,\pi)$: $\dfrac2\pi\int_0^\pi f^2dx=\dfrac{a_0^2}{2}+\sum a_n^2$.
It is used to sum series such as $\sum 1/n^2$ or $\sum 1/n^6$.

Example. $f=x$ on $(-\pi,\pi)$ has $b_n=\dfrac{2(-1)^{n+1}}{n}$ and $\dfrac1\pi\int_{-\pi}^{\pi}x^2dx=\dfrac{2\pi^2}{3}$, so Parseval gives $4\sum\dfrac1{n^2}=\dfrac{2\pi^2}{3}$, that is $$\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\dots=\frac{\pi^2}{6}$$ 4. For general period $2l$, use $\dfrac1l\int_{-l}^{l}f^2dx=\dfrac{a_0^2}{2}+\sum(a_n^2+b_n^2)$. 5. Parseval follows by squaring the series, integrating term by term and using orthogonality of $\sin nx$ and $\cos nx$.

Last-minute revision

Geometric series converges to $a/(1-r)$ when $|r|<1$; $\sum1/4^n=1/3$.
p-series converges only for $p>1$.
Ratio test: limit below 1 converges, above 1 diverges.
Taylor: $f(x)=\sum f^{(n)}(a)(x-a)^n/n!$.
$\sin x$ about $\pi/2$ is $1-(x-\pi/2)^2/2!+(x-\pi/2)^4/4!-\dots$
$\log(1+x)$ is valid for $-1<x\le1$.
Fourier coefficients use $1/\pi$ over $(-\pi,\pi)$; even gives $b_n=0$, odd gives $a_n=0$.
Half range series use $2/\pi$ over $(0,\pi)$.
$x(\pi-x)$ sine series has $b_n=8/(\pi n^3)$ for odd $n$ and gives $\pi^3/32$.
Parseval: $\frac1\pi\int f^2=a_0^2/2+\sum(a_n^2+b_n^2)$.

Memory hooks

Ratio less than one, series is done (converges).
Even function keeps cos, odd function keeps sin.
Half range means double the integral and use $2/\pi$.
Parseval: square the coefficients, add them, match the mean square.

Coverage checklist

Convergence of sequence and series: Dec 2024 series $\sum1/4^n$.
tests for convergence: no recent questions.
Power series: no recent questions.
Taylor's series: Dec 2024 sin x about $\pi/2$.
series for exponential, trigonometric and logarithm functions: no recent questions.
Fourier series: Half range sine and cosine series: Jun 2025 x+x^2, Dec 2024 e^x, Jun 2025 x(pi-x).
Parseval’s theorem: no recent questions.

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Mathematics I (BT-102) - Unit 3 Short Notes

How unit 3 is examined

Convergence of sequence and series

Tests for convergence

Power series

Taylor's series

Series for exponential, trigonometric and logarithm functions

Fourier series: Half range sine and cosine series

Parseval's theorem

Last-minute revision

Memory hooks

Coverage checklist