Infinite Series – NCERT Class 11 Mathematics Appendix 1 – Binomial Theorem, Infinite Geometric Series, Exponential and Logarithmic Series Studies infinite sequences and series including binomial theorem for any index, infinite geometric series with convergence criteria, exponential series defining number e e, its properties, and logarithmic series. Discusses general term derivations, convergence conditions, applications, and illustrative examples from calculus and algebra. Updated: just now
Categories: NCERT, Class XI, Mathematics, Infinite Series, Binomial Theorem, Geometric Series, Exponential Series, Logarithmic Series, Appendix 1
Tags: Infinite Series, Binomial Theorem, Infinite Geometric Series, Exponential Series, Logarithmic Series, Sigma Notation, Series Convergence, Number e e, Calculus Applications, NCERT Class 11, Mathematics, Appendix 1
Infinite Series: NCERT Class 11 Appendix 1 - Ultimate Study Guide, Notes, Theorems, Examples 2025
Full Appendix Summary & Detailed Notes
Binomial Theorem for Any Index
Infinite Geometric Series
Exponential Series
Logarithmic Series
Questions & Answers
Quick Revision Notes & Mnemonics
Formulas & Theorems
Full Appendix Summary & Detailed Notes - Infinite Series Class 11 NCERT
Overview & Key Concepts
Appendix Goal : Special infinite series: Binomial (any index), geometric, exponential, logarithmic. Focus: Expansions, sums, applications. Exam Focus: Theorems, examples, |x|<1 condition. 2025 Updates: Euler's e emphasis. Fun Fact: e ≈ 2.718 from series. Core Idea: Infinite sums for functions. Real-World: Approximations, calculus. Ties: Ch9 sequences. Expanded: From finite to infinite.Wider Scope : Theorems without proofs; examples illustrate. Covers A.1.1 to A.1.5.Expanded Content : Theorems, remarks, examples step-by-step.
A.1.1 Introduction
Infinite sequence sum as series $$ \sum_{k=1}^{\infty} a_k $$. Study special types.
A.1.2 Binomial Theorem
General: $$ (1+x)^m = 1 + mx + \frac{m(m-1)}{2!}x^2 + \cdots $$ for |x|<1, m any. Remark: Condition necessary. General term: $$ T_{r+1} = \frac{m(m-1)\cdots(m-r+1)}{r!} x^r $$. For (a+b)^m: Valid |b/a|<1.
A.1.3 Infinite Geometric Series
Sum $$ S = \frac{a}{1-r} $$ if |r|<1. Ex: $$ \frac{1}{3} + \frac{2}{9} + \cdots = \frac{3}{2} $$. Behavior: r^n →0 as n→∞.
A.1.4 Exponential Series
e = $$ 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots $$ , 2
A.1.5 Logarithmic Series
$$ \log_e(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots $$ , |x|<1. Note: At x=1, log2 = $$ 1 - \frac{1}{2} + \frac{1}{3} - \cdots $$ . Ex: Proof for roots α,β.
Summary
Tools for expansions/sums. Master: Conditions, general terms. Apps: Approximations. Mantra: |r|<1 converges.
Why This Guide Stands Out
Theorems boxed, examples solved, series approximations, free 2025 with MathJax.
Key Themes & Tips
Aspects : From binomial to logs via series.Tip: Check |x|<1; use factorial for exp.
Exam Case Studies
Expand (1+x)^{-1/2}; sum geometric; coeff in e^x.
Project & Group Ideas
Compute e via Python series sum.
Apps: Series calculator for approximations.
Binomial Theorem for Any Index - Theorem & Examples
Theorem: Binomial Expansion for Non-Integer m
$$ (1+x)^m = 1 + mx + \frac{m(m-1)}{1 \cdot 2} x^2 + \frac{m(m-1)(m-2)}{1 \cdot 2 \cdot 3} x^3 + \cdots $$ holds whenever $$ |x| < 1 $$.
General Term: $$ T_{r+1} = \frac{m(m-1)\cdots(m-r+1)}{r!} x^r $$
Condition: |x|<1 necessary for convergence (e.g., x=-2, m=-2 diverges).
For (a+b)^m: Valid when $$ \left| \frac{b}{a} \right| < 1 $$, expansion $$ a^m \left(1 + \frac{b}{a}\right)^m $$.
Example 1: Expand $$ \left(1 - \frac{x}{2}\right)^{-1/2} $$, |x|<2
Step 1: Write as $$ \left(1 + \left(-\frac{x}{2}\right)\right)^{-1/2} $$
Step 2: m=-1/2, x=-x/2
Step 3: Terms: 1 + (1/2)(x/2) + (1/2)(3/2)/2! (x/2)^2 + ...
Result: $$ 1 + \frac{x}{4} + \frac{3x^2}{32} + \cdots $$
Particular Cases (|x|<1):
$$ (1+x)^{-1} = 1 - x + x^2 - x^3 + \cdots $$
$$ (1-x)^{-1} = 1 + x + x^2 + x^3 + \cdots $$
$$ (1+x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \cdots $$
$$ (1-x)^{-2} = 1 + 2x + 3x^2 + 4x^3 + \cdots $$
Tip: Infinite terms; binomial coeff generalized.
Infinite Geometric Series - Theorem & Examples
Theorem: Sum of Infinite GP
For GP a, ar, ar^2, ..., if $$ |r| < 1 $$, then $$ S_\infty = \frac{a}{1-r} $$ (as $$ r^n \to 0 $$ as $$ n \to \infty $$).
Finite Recall: $$ S_n = \frac{a(1-r^n)}{1-r} $$ → $$ \frac{a}{1-r} $$ when |r|<1.
Example 2: Sum $$ -\frac{5}{4}, -\frac{5}{16}, -\frac{5}{64}, \cdots $$
Step 1: a = -5/4, r = -1/4, |r|=1/4<1
Step 2: $$ S = \frac{-5/4}{1 - (-1/4)} = \frac{-5/4}{5/4} = -1 $$
Illustrations:
$$ 1 + \frac{1}{2} + \frac{1}{4} + \cdots = 2 $$
$$ 1 - \frac{1}{2} + \frac{1}{4} - \cdots = \frac{2}{3} $$
Tip: Check |r|<1 for convergence; table shows r^n →0.
Exponential Series - Theorem & Examples
Theorem: Definition of e and Exponential Series
$$ e = \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots $$ , 2 < e < 3.
General: $$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} $$
Bounds Proof: Partial sums <3, >2 via inequalities like $$ \frac{1}{n!} < \frac{1}{2^{n-1}} $$ for n>2.
Example 3: Coeff of x^2 in e^{2x+3}
Step 1: e^{2x+3} = e^3 e^{2x} = e^3 \sum \frac{(2x)^n}{n!}
Step 2: Term for n=2: e^3 \frac{4x^2}{2!} = 2 e^3 x^2
Result: 2e^3
Example 4: e^2 to 1 decimal
Step 1: Sum first 7 terms >7.355
Step 2: Upper bound <7.4
Result: 7.4
Tip: Factorials grow fast; approx by partial sums.
Logarithmic Series - Theorem & Examples
Theorem: Logarithmic Expansion
If $$ |x| < 1 $$, $$ \log_e (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots $$
Note: Valid at x=1: $$ \log_e 2 = 1 - \frac{1}{2} + \frac{1}{3} - \cdots $$
Example 5: Prove for roots α,β of x^2 - px + q=0
Step 1: RHS = $$ \log_e (1 + \alpha x) + \log_e (1 + \beta x) = \log_e [(1 + \alpha x)(1 + \beta x)] $$
Step 2: (1 + αx)(1 + βx) = 1 + (α+β)x + αβ x^2 = 1 + p x + q x^2
Step 3: log(1 + p x + q x^2) = LHS (by assumption |αx|<1, |βx|<1)
Tip: Alternating; use for log approximations.
20 Questions & Answers - NCERT Based (Short & Detailed)
Part A: 10 short (1-2 marks), Part B: 10 detailed (4-6 marks). From examples/variations.
Part A: Short Questions (10 Qs)
1. Condition for binomial series?
Answer: $$ |x| < 1 $$
2. Infinite GP sum formula?
Answer: $$ \frac{a}{1-r} $$ , |r|<1
10. Coeff x^2 in e^x?
Answer: $$ \frac{1}{2} $$
Part B: Detailed Questions (10 Qs)
1. Expand (1+x)^{-1} up to x^3.
Detailed Answer:
m=-1: 1 + (-1)x + \frac{-1(-2)}{2!}x^2 + \frac{-1(-2)(-3)}{3!}x^3 + ...
= 1 - x + x^2 - x^3 + ...
10. Prove log(1+x) series for x=1/2.
Detailed Answer:
Substitute x=1/2: 1/2 - (1/2)^2/2 + (1/2)^3/3 - ...
Approx value.
Tip: Show steps for expansions.
Quick Revision Notes & Mnemonics
Binomial
$$ (1+x)^m $$ , |x|<1; T_r = \frac{m^{\underline{r}}}{r!} x^r
Mnemonic: "Binom Any Index Converge Absolute" (BAICA)
Geometric
S= a/(1-r), |r|<1
Mnemonic: "Ratio Less One Sum" (RLOS)
Exponential
e^x = \sum x^n / n!
Mnemonic: "Euler Expands Factorials" (EEF)
Log
log(1+x)= \sum (-1)^{n+1} x^n / n , |x|<1
Mnemonic: "Log Alternates Divide N" (LADN)
Overall Mnemonic: "Big Geo Exp Log" (BGEL). Flashcards for formulas.
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