Question:
The coefficient of $x^n$ in the expansion of $\log_a (1 + x)$ is
The coefficient of $x^n$ in the expansion of $\log_a (1 + x)$ is
Updated On: Jun 18, 2022
- $\frac{(-1)^{n - 1}}{n}$
- $\frac{(-1)^{n - 1}}{n} \log_a \, e $
- $\frac{(-1)^{n - 1}}{n} \log_e \, a $
- $\frac{(-1)^{n }}{n} \log_a \, e $
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
$\log_{a} \left(1+x\right) = \log_{e} \left(1+x\right)\log_{a}e$
$ = \log_{a}e \left[\displaystyle\sum_{n=1}^{\infty}\left(-1\right)^{n-1} \frac{X^{n}}{n}\right]$
So, the coefficient of $x^n$ in $\log_a ( 1 + x)$ is
$ \frac{\left(-1\right)^{n- 1}}{n} \log_{a}e $
$ = \log_{a}e \left[\displaystyle\sum_{n=1}^{\infty}\left(-1\right)^{n-1} \frac{X^{n}}{n}\right]$
So, the coefficient of $x^n$ in $\log_a ( 1 + x)$ is
$ \frac{\left(-1\right)^{n- 1}}{n} \log_{a}e $
Was this answer helpful?
0
0
Top Questions on binomial expansion formula
- Coefficient of x2012 in (1-x)2008(1+x+x²)2007 is equal to ___
- JEE Main - 2024
- Mathematics
- binomial expansion formula
- Number of integral terms in the expansion of \( \left( \sqrt{7} z + \frac{1}{6 \sqrt{z}} \right)^{824} \) is equal to ______.
- JEE Main - 2024
- Mathematics
- binomial expansion formula
- In the expansion of \((1 + x)(1 - x^2) (1 + \frac 3x + \frac {3}{x^2}+ \frac {1}{x^3})^5\) the sum of coefficients of \(x^3\) and \(x^{-13}\) is
- JEE Main - 2024
- Mathematics
- binomial expansion formula
- The absolute difference of the coefficient of x7 and x9 in the expansion of \((\frac{2x+1}{2x})^{11}\) is?
- JEE Main - 2023
- Mathematics
- binomial expansion formula
The coefficient of x7 in (1 – 2x + x3)10 is?
- JEE Main - 2023
- Mathematics
- binomial expansion formula
View More Questions
Questions Asked in VITEEE exam
- Two identical blocks A and B, each of mass 'm' resting on a smooth floor are connected by a light spring of natural length L and spring constant K, with the spring at its natural length. A third identical block 'C' (mass m) moving with a speed v along the line joining A and B collides with A. the maximum compression in the spring is
- VITEEE - 2023
- work, energy and power
- Assertion :
The binding energy of the nucleus increases with the increase in atomic number.
Reason :
Heavier elements have a greater number of non-radioactive isotopes than radioactive isotopes. - The half-life of radioactive radon is 3.8 days. The time at the end of which \(\frac{1}{20}th\) of the Radon sample will remain undecayed is (given log10e=0.4343)
- Which element has more electron gain enthalpy among chalcogen group?
- VITEEE - 2022
- sulphur
- Coin is tossed till heads is obtained what is expectation of no. of coin tosses
- VITEEE - 2022
- Random Variables
View More Questions
Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .