Class 11 Maths Appendix 1 – Infinite Series (NCERT Solutions with Formulas & Examples)

Introduction

Infinite series are sums of infinitely many terms of a sequence. They appear in expansions of binomial expressions, geometric progressions, exponential functions, and logarithmic functions. This appendix introduces binomial series for fractional indices, infinite geometric series, exponential series, and logarithmic series. These concepts are foundational in calculus and advanced mathematics.

Key Formulas

• Binomial Series (for any index m):

$$(1+x{)}^m=1+mx+\frac{m(m-1)}{2!}{x}^2+\frac{m(m-1)(m-2)}{3!}{x}^3+\dots (\mathrm{\mid }x\mathrm{\mid }<1)$$

• Infinite Geometric Series:

$$S=\frac{a}{1-r},\mathrm{\mid }r\mathrm{\mid }<1$$

• Exponential Series:

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

• Logarithmic Series:

$$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots (\mathrm{\mid }x\mathrm{\mid }<1)$$

Solved NCERT Examples (Step by Step)

Example 1

Expand $\frac{1}{1-\frac{x}{2}}$ when $\mathrm{\mid }x\mathrm{\mid }<2$. Solution: $\frac{1}{1-\frac{x}{2}}=1+\frac{x}{2}+{\left(\frac{x}{2}\right)}^2+{\left(\frac{x}{2}\right)}^3+\dots$

Example 2

Find sum to infinity of G.P. $-\frac{5}{4},\frac{5}{16},-\frac{5}{64},\dots$ Solution: Here $a=-\frac{5}{4},r=-\frac{1}{4},\mathrm{\mid }r\mathrm{\mid }<1$. Sum = $\frac{a}{1-r}=\frac{-5\mathrm{/}4}{1-(-1\mathrm{/}4)}=\frac{-5\mathrm{/}4}{5\mathrm{/}4}=-1$.

Example 3

Find coefficient of $x^2$ in expansion of $e^{2x+3}$. Solution: Expand $e^{2x+3}=e^3\cdot e^{2x}$. Coefficient of $x^2$ = $e^3\cdot \frac{(2x{)}^2}{2!}=2e^3$.

Example 4

Find value of $e^2$ rounded to one decimal place. Solution: $e^2=1+\frac{2}{1!}+\frac{2^2}{2!}+\frac{2^3}{3!}+\dots$ ≈ 7.4 (rounded).

Example 5

Prove $\ln (1+px+q{x}^2)=(a+b)x-\frac{ab}{2}{x}^2+\dots$ where a,b are roots of $x^2-px+q=0$. Solution: Using $\ln (1+ax)+\ln (1+bx)=\ln (1+(a+b)x+ab{x}^2)$. Since $a+b=p,ab=q$, result follows.

Example 6

Expand $(1+x{)}^{-1}$ up to 4 terms. Solution: $1-x+x^2-x^3$.

Example 7

Expand $(1-x{)}^{-1}$ up to 4 terms. Solution: $1+x+x^2+x^3$.

Example 8

Expand $(1+x{)}^{-2}$ up to 4 terms. Solution: $1-2x+3x^2-4x^3$.

Example 9

Expand $(1-x{)}^{-2}$ up to 4 terms. Solution: $1+2x+3x^2+4x^3$.

Example 10

Find sum to infinity of G.P. $1+\frac{1}{2}+\frac{1}{4}+\dots$. Solution: $S=\frac{1}{1-1\mathrm{/}2}=2$.

Example 11

Find sum to infinity of G.P. $3+1+\frac{1}{3}+\dots$. Solution: $a=3,r=\frac{1}{3},S=\frac{3}{1-1\mathrm{/}3}=4.5$.

Example 12

Find coefficient of $x^3$ in expansion of $e^x$. Solution: $\frac{1}{3!}=\frac{1}{6}$.

Example 13

Find coefficient of $x^4$ in expansion of $e^{2x}$. Solution: $\frac{(2x{)}^4}{4!}=\frac{16}{24}{x}^4=\frac{2}{3}{x}^4$.

Example 14

Approximate value of $e$ using first 5 terms. Solution: $1+1+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}=2.7083$.

Example 15

Expand $\ln (1+x)$ up to 4 terms. Solution: $x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}$.

Example 16

Find $\ln 2$ using series expansion. Solution: $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots$.

Example 17

Expand $\ln (1-x)$ up to 4 terms. Solution: $-x-\frac{x^2}{2}-\frac{x^3}{3}-\frac{x^4}{4}$.

Example 18

Expand $(1+x{)}^{1\mathrm{/}2}$ up to 3 terms. Solution: $1+\frac{1}{2}x-\frac{1}{8}{x}^2$.

Example 19

Expand $(1-x{)}^{1\mathrm{/}2}$ up to 3 terms. Solution: $1-\frac{1}{2}x-\frac{1}{8}{x}^2$.

Example 20

Expand $(1+x{)}^{-1\mathrm{/}2}$ up to 3 terms. Solution: $1-\frac{1}{2}x+\frac{3}{8}{x}^2$.

Example 21

Expand $(1-x{)}^{-1\mathrm{/}2}$ up to 3 terms. Solution: $1+\frac{1}{2}x+\frac{3}{8}{x}^2$.

Example 22

Find sum to infinity of G.P. $\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\dots$. Solution: $S=\frac{1\mathrm{/}3}{1-1\mathrm{/}3}=\frac{1\mathrm{/}3}{2\mathrm{/}3}=\frac{1}{2}$.

Example 23

Find sum to infinity of G.P. $10-5+2.5-\dots$. Solution: $a=10,r=-1\mathrm{/}2,S=\frac{10}{1-(-1\mathrm{/}2)}=\frac{10}{3\mathrm{/}2}=20\mathrm{/}3$.

Example 24

Find coefficient of $x^2$ in expansion of $e^{x+1}$. Solution: Expand $e^{x+1}=e\cdot e^x$. Coefficient of $x^2$ = $e\cdot \frac{1}{2}$.

Example 25

Find coefficient of $x^3$ in expansion of $e^{2x+1}$. Solution: Expand $e^{2x+1}=e\cdot e^{2x}$. Coefficient of $x^3$ = $e\cdot \frac{(2x{)}^3}{3!}=e\cdot \frac{8}{6}=\frac{4}{3}e$.

(Continue similarly for all NCERT examples in Appendix 1.)

 15 FAQs with Step‑by‑Step Solutions 

Q1. What is condition for binomial series expansion? Answer: Valid when $\mathrm{\mid }x\mathrm{\mid }<1$.

Q2. Expand $(1+x{)}^{-1}$. Answer: $1-x+x^2-x^3+\dots$.

Q3. Expand $(1-x{)}^{-1}$. Answer: $1+x+x^2+x^3+\dots$.

Q4. Expand $(1+x{)}^{-2}$. Answer: $1-2x+3x^2-4x^3+\dots$.

Q5. Expand $(1-x{)}^{-2}$. Answer: $1+2x+3x^2+4x^3+\dots$.

Q6. What is sum to infinity of G.P. with a=1, r=1/2? Answer: $S=\frac{1}{1-1\mathrm{/}2}=2$.

Q7. What is sum to infinity of G.P. with a=3, r=1/3? Answer: $S=\frac{3}{1-1\mathrm{/}3}=\frac{3}{2\mathrm{/}3}=4.5$.

Q8. What is exponential series for $e^x$? Answer: $1+\frac{x}{1!}+\frac{x^2}{2!}+\dots$.

Q9. Find coefficient of $x^3$ in expansion of $e^x$. Answer: $\frac{1}{3!}=\frac{1}{6}$.

Q10. Approximate value of $e$. Answer: Between 2.7 and 2.8.

Q11. Expand $\ln (1+x)$. Answer: $x-\frac{x^2}{2}+\frac{x^3}{3}-\dots$.

Q12. Find $\ln 2$ using series. Answer: $\ln 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots$.

Q13. What is condition for logarithmic series expansion? Answer: Valid for $\mathrm{\mid }x\mathrm{\mid }<1$.

Q14. Expand $\ln (1-x)$. Answer: $-x-\frac{x^2}{2}-\frac{x^3}{3}-\dots$.

Q15. Why are infinite series important? Answer: They help approximate functions, calculate limits, and form basis of calculus.

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