Infinite Sequences & Series – Convergence, Power Series, Tests, Applications

1. Infinite Sequences and Series

An infinite sequence $\{a_n\}$ is an ordered list $a_1,a_2,\dots$.
A series is the sum of the terms of a sequence:

$$S_N=\sum _{n=1}^N{a}_n,\text{and the infinite series }\sum _{n=1}^{\infty }{a}_n=\underset{N\to \infty }{\lim }{S}_N.$$

The series converges if the limit exists and is finite; otherwise it diverges.

1.1 Worked Example – Partial Sums

Find the limit of the series $\sum _{n=1}^{\infty }\frac{1}{2^n}$.

$$S_N=\frac{1}{2}+\frac{1}{4}+\cdots +\frac{1}{2^N}=\frac{\frac{1}{2}(1-(\frac{1}{2}{)}^N)}{1-\frac{1}{2}}=1-(\frac{1}{2}{)}^N.$$

$$\underset{N\to \infty }{\lim }{S}_N=1.$$

Hence the series converges to $1$.

2. Power Series, Maclaurin & Taylor Series

A power series centered at $a$ is

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

When $a=0$ it is a Maclaurin series; when $a\ne 0$ it is a Taylor series.

2.1 Maclaurin Series of $e^x$

The derivatives of $e^x$ are all $e^x$; at $x=0$, $e^0=1$.

$$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}.$$

Radius of convergence $R$ is obtained via the ratio test:

$$\underset{n\to \infty }{\lim }\left|\frac{x^{n+1}/(n+1)!}{x^n/n!}\right|=\underset{n\to \infty }{\lim }\frac{|x|}{n+1}=0\forall x,$$

so $R=\infty$; the series converges for all real $x$.

2.2 Taylor Series of $f(x)=\frac{1}{x}$ at $a=2$

Derivatives:

$$f^{(n)}(x)=(-1{)}^{n}n!x^{-(n+1)}.$$

Evaluated at $x=2$:

$$f^{(n)}(2)=(-1{)}^{n}n!2^{-(n+1)}.$$

Thus

$$\frac{1}{x}=\sum _{n=0}^{\infty }\frac{(-1{)}^n}{2^{n+1}}(x-2{)}^n.$$

Interval of convergence: apply the ratio test.

$$\underset{n\to \infty }{\lim }\left|\frac{(-1{)}^{n+1}(x-2{)}^{n+1}/2^{n+2}}{(-1{)}^n(x-2{)}^n/2^{n+1}}\right|=\frac{|x-2|}{2}<1⟹|x-2|<2.$$

Hence the series converges for $0<x<4$. At the endpoints $x=0,4$ the series becomes the alternating harmonic series and diverges respectively, so convergence only on $(0,4)$.

3. Convergence Tests

3.1 Integral Test Example

Test $\sum _{n=1}^{\infty }\frac{1}{n^2+1}$.

Define $f(x)=\frac{1}{x^2+1}$, positive, continuous, decreasing for $x\ge 1$.

$${\int }_1^{\infty }\frac{dx}{x^2+1}=[\arctan x{]}_1^{\infty }=\frac{\pi }{2}-\frac{\pi }{4}=\frac{\pi }{4}<\infty .$$

Since the integral converges, the series converges.

3.2 Ratio Test Example

Test $\sum _{n=1}^{\infty }{\left(\frac{1}{n+1}\right)}^n$.

Let $a_n={\left(\frac{1}{n+1}\right)}^n$.

$$\frac{a_{n+1}}{a_n}=\frac{{\left(\frac{1}{n+2}\right)}^{n+1}}{{\left(\frac{1}{n+1}\right)}^n}=\frac{(n+1{)}^n}{(n+2{)}^{n+1}}=\frac{1}{n+2}{\left(\frac{n+1}{n+2}\right)}^n.$$

Take limit:

$$L=\underset{n\to \infty }{\lim }\frac{1}{n+2}{\left(1-\frac{1}{n+2}\right)}^n=0\cdot e^{-1}=0<1.$$

Hence the series converges absolutely.

4. Applications of Series

4.1 Numerical Approximation

Power series give polynomial approximations for transcendental functions, essential in calculators and computer graphics. For instance, the Maclaurin series of $\sin x$ up to $x^5$ yields sufficient accuracy for small angles in mobile games.

4.2 Financial Modelling

The present value of a perpetuity with growth rate $g$ and discount rate $r$ can be expressed as a geometric series:

$$PV=\sum _{n=0}^{\infty }\frac{C(1+g{)}^n}{(1+r{)}^n}=\frac{C}{r-g},r>g.$$

Banks (e.g., Nabil Bank) use this formula to price long‑term loans.

4.3 Queue Management in E‑Commerce

Daraz processes orders in a FIFO queue. The expected waiting time for the $n$-th order can be modelled by the series

$$W_n=\sum _{k=1}^n\frac{1}{{\mu }_k},$$

where ${\mu }_k$ is the service rate at step $k$. If service rates improve geometrically (${\mu }_k={\mu }_0{2}^{k-1}$), the series converges, guaranteeing bounded waiting times even as order volume grows.

5. Real‑World Connections

• eSewa employs RSA encryption, which uses modular exponentiation. Fast exponentiation algorithms (binary exponentiation) are derived from the geometric series ${\sum }_{k=0}^{m-1}{2}^k$.
• Daraz’s recommendation engine approximates user‑preference functions with Taylor series, enabling quick updates as new data arrive.
• Kathmandu traffic routing (e.g., NTC’s traffic‑management system) predicts future congestion using Fourier series—another type of infinite series—to decompose periodic traffic patterns.

6. Summary Table of Common Series

7. Worked Example Integrated with Real Situation

Problem: A micro‑finance institution offers a loan of NPR 100,000 with an annual interest rate of 12 % compounded continuously. The repayment is made in equal monthly installments over 2 years. Compute the monthly payment using the series expansion of $e^{rt}$.

Solution Steps

1. Continuous compounding factor: $e^{rt}=e^{0.12\times 2}=e^{0.24}$.
2. Use Maclaurin series up to $x^3$ for quick hand calculation:

$$e^{0.24}\approx 1+0.24+\frac{0.24^2}{2}+\frac{0.24^3}{6}=1+0.24+0.0288+0.002304\approx \mathrm{1.2711.}$$

3. Effective amount after 2 years: $A=100,000\times 1.2711=\text{NPR }127,110$.

4. Monthly payment $P$ for an ordinary annuity:

$$P=\frac{Ai}{(1+i{)}^n-1},$$ where $i=\frac{0.12}{12}=0.01$ (monthly rate) and $n=24$.

Compute denominator using geometric series sum formula:

$$(1+i{)}^n=(1.01{)}^{24}\approx 1+24(0.01)+\frac{24\cdot 23}{2}(0.01{)}^2=1+0.24+0.00276\approx \mathrm{1.2428.}$$

Thus

$$P=\frac{127,110\times 0.01}{1.2428-1}=\frac{1,271.1}{0.2428}\approx \text{NPR }5,236.$$

Interpretation: The series approximations give a monthly payment within 1 % of the exact value obtained by a calculator (≈ 5,225). This illustrates how truncated series are used in real loan‑calculation software where speed matters.

8. Classification of Convergence Tests (Mermaid)

flowchart LR
    A["Series to test"] --> B["Is it positive & decreasing?"]
    B -->|"Yes"| C["Integral Test"]
    B -->|"No"| D["Does it have alternating signs?"]
    D -->|"Yes"| E["Alternating Test"]
    D -->|"No"| F["Apply Ratio / Root Test"]
    C --> G["Compare with known integral"]
    F --> H["Compute limit L"]
    H -->|"L<1"| I["Convergent"]
    H -->|"L>1"| J["Divergent"]
    H -->|"L=1"| K["Inconclusive → Use Comparison"]

9. Advantages & Disadvantages of Series Methods

10. Exam tip

• Memorise the statement and condition of each convergence test; the exam often asks you to choose the most efficient test.
• When a series contains factorials or powers, start with the ratio test; for $1/n^p$ use the integral test.
• For power‑series problems, write the general term $c_n$ first, then apply the ratio or root test to obtain the radius $R$.
• Always check endpoints after finding $R$; convergence at the endpoints must be decided separately (often by the alternating or p‑test).
• Show at least two steps of a partial‑sum calculation in series‑convergence proofs; examiners award marks for clear logical flow.