Write the first 4 terms of the logarithmic series log 1 + 3 x 1 - 3 x . Find the intervals on which the expansions are valid.

Given that log 1 + 3 x 1 - 3 x We know that log 1 + x 1 - x = log ( 1 + x ) - log ( 1 - x ) = 2 x + x 3 3 + x 5 5 + ⋯ · , where x < 1 ⇒ log 1 + 3 x 1 - 3 x = 2 3 x + ( 3 x ) 3 3 + ( 3 x ) 5 5 + ( 3 x ) 7 7 + ⋯ ⇒ 2 3 x + 9 x 3 + 24 3 5 x 2 + 287 x 7 7 + ⋯ , | x | < 1 3 , | 3 x | < 1

Write the first 4 terms of the logarithmic series log 1 - 2 x 1 + 2 x . Find the intervals on which the expansion is valid.

Given that log 1 - 2 x 1 + 2 x We know that log 1 - x 1 + x = log ( 1 - x ) - log ( 1 + x ) = - 2 x - 2 x 3 3 - 2 x 5 5 - 2 x 7 7 ⇒ - 2 x + x 3 3 + x 5 5 + x 5 7 + ⋯ log 1 - 2 x 1 + 2 x ⇒ - 2 2 x + ( 2 x ) 3 3 + ( 2 x ) 5 5 + ( 2 x ) 7 7 + ⋯ ⇒ - 2 2 x + 8 x 3 3 + 32 x 5 5 + 128 x 7 7 + . . . . . where | x | < 1 2 , | 2 x | < 1

If y = x + x 2 2 + x 3 3 + x 4 4 + ⋯ , then show that x = y - y 2 2 ! + y 3 3 ! - y 4 4 ! + ⋯

Given that y = x + x 2 2 + x 3 3 + x 4 4 + ⋯ ⇒ - y = - x + x 2 2 + x 3 3 + x 4 4 + ⋯ = - x - x 2 2 - x 3 3 - x 4 4 … ⇒ - y = log ( 1 - x ) ⇒ e - y = 1 - x ⇒ x = 1 - e - y We know that e - x = 1 - x 1 ! + x 2 2 ! - x 3 3 ! + ⋯ x = 1 - 1 - y 1 ! + y 2 2 ! - y 3 3 ! + y 4 4 ! - ⋯ ⇒ x = y - y 2 2 ! + y 3 3 ! - y 4 4 ! + ⋯ Hence proved

If p − q is small compared to either p or q , then show that p q n ≃ ( n + 1 ) p + ( n - 1 ) q ( n - 1 ) p + ( n + 1 ) q . Hence find 15 16 8 .

Given that L H S = n p q ⇒ ( p q ) 1 n = 2 p 2 q 1 n = p + p + q - q q + q + p - p 1 n ⇒ ( p + q ) + ( p - q ) ( p + q ) - ( p - q ) 1 n Dividing with p + q on both numerator and denominator, ⇒ p + q p + q + p - q p + q p + q p + q - p - q p + q 1 / n ⇒ 1 + p - q p + q 1 n 1 - p - q p + q 1 n ⇒ 1 + 1 n p - q p + q 1 - 1 n p - q p + q So, p q n = p ( n + 1 ) + q ( n - 1 ) p ( n - 1 ) + q ( n + 1 ) , Hence proved 15 16 8 ⇒ 15 ( 8 + 1 ) + 16 ( 8 - 1 ) 15 ( 8 - 1 ) + 16 ( 8 + 1 ) ⇒ 0 . 99196

Find the value of ∑ n = 1 ∞ 1 2 n - 1 1 9 n - 1 + 1 9 2 n - 1

Given that ∑ n = 1 ∞ 1 ( 2 n - 1 ) · 1 9 n - 1 + ∑ n = 1 1 1 ( 2 n - 1 ) · 1 9 2 n - 1 ⇒ 1 2 ( 1 ) - 1 · 1 9 1 - 1 + 1 2 ( 2 ) - 1 · 1 9 2 - 1 + … + 1 2 ( 1 ) - 1 · 1 9 2 - 1 + 1 2 2 - 1 1 9 2 2 - 1 + … ⇒ 1 1 9 0 + 1 3 · 1 9 1 + … + 1 1 · 1 9 1 + 1 3 1 9 3 + … We know log 1 + x 1 - x = 2 x + x 3 3 + x 5 5 + ⋯ ⇒ 1 + 1 9 3 + 1 9 2 5 + … + 1 9 1 + 1 9 3 3 + … ⇒ 1 + 1 3 2 3 + 1 3 4 5 + … + 1 9 1 + 1 9 3 3 + … Multiply first term with 1 3 ⇒ 3 1 3 + 1 3 3 3 + 1 3 5 3 + ⋯ · + 1 9 + 1 9 3 3 + 1 9 5 5 + ⋯ ⇒ 3 1 2 log 1 + 1 / 3 1 - 1 / 3 + 1 2 log 1 + 1 9 1 - 1 9 ⇒ 1 2 log 2 3 · 5 4 ⇒ 1 2 log e 10

E M B I B E

Mathematics Standard 11 Vol I>Chapter 5 - Binomial Theorem, Sequences and Series>Exercise

Tamil Nadu Board Solutions for Chapter: Binomial Theorem, Sequences and Series, Exercise 4: Exercise

Author:Tamil Nadu Board

Tamil Nadu Board Mathematics Solutions for Exercise - Tamil Nadu Board Solutions for Chapter: Binomial Theorem, Sequences and Series, Exercise 4: Exercise

Attempt the free practice questions on Chapter 5: Binomial Theorem, Sequences and Series, Exercise 4: Exercise with hints and solutions to strengthen your understanding. Mathematics Standard 11 Vol I solutions are prepared by Experienced Embibe Experts.

Questions from Tamil Nadu Board Solutions for Chapter: Binomial Theorem, Sequences and Series, Exercise 4: Exercise with Hints & Solutions

EASY

11th Tamil Nadu Board

IMPORTANT

Mathematics Standard 11 Vol I>Chapter 5 - Binomial Theorem, Sequences and Series>Exercise>Q 6

Write the first $4$ terms of the logarithmic series $\log (1 + 4 x)$ .Find the intervals on which the expansions are valid.

EASY

11th Tamil Nadu Board

IMPORTANT

Mathematics Standard 11 Vol I>Chapter 5 - Binomial Theorem, Sequences and Series>Exercise>Q 6

Write the first $4$ terms of the logarithmic series $\log (1 - 2 x)$ .Find the intervals on which the expansions are valid.

EASY

11th Tamil Nadu Board

IMPORTANT

Mathematics Standard 11 Vol I>Chapter 5 - Binomial Theorem, Sequences and Series>Exercise>Q 6

Write the first $4$ terms of the logarithmic series $\log (\frac{1 + 3 x}{1 - 3 x})$ . Find the intervals on which the expansions are valid.

EASY

11th Tamil Nadu Board

IMPORTANT

Mathematics Standard 11 Vol I>Chapter 5 - Binomial Theorem, Sequences and Series>Exercise>Q 6

Write the first $4$ terms of the logarithmic series $\log (\frac{1 - 2 x}{1 + 2 x})$ . Find the intervals on which the expansion is valid.

EASY

11th Tamil Nadu Board

IMPORTANT

Mathematics Standard 11 Vol I>Chapter 5 - Binomial Theorem, Sequences and Series>Exercise>Q 7

If $y = x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + \frac{x^{4}}{4} + \dots$ , then show that $x = y - \frac{y^{2}}{2!} + \frac{y^{3}}{3!} - \frac{y^{4}}{4!} + \dots$

EASY

11th Tamil Nadu Board

IMPORTANT

Mathematics Standard 11 Vol I>Chapter 5 - Binomial Theorem, Sequences and Series>Exercise>Q 8

If $p - q$ is small compared to either $p$ or $q$ , then show that $\sqrt[n]{\frac{p}{q}} ≃ \frac{(n + 1) p + (n - 1) q}{(n - 1) p + (n + 1) q}$ . Hence find $\sqrt[8]{\frac{15}{16}}$ .