Taylor Series Expansion of sin(z)
The Taylor series expansion of the function $f(z) = \sin(z)$ around $z = 0$ (also known as the Maclaurin series) is given by:
$ \sin(z) = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \dots $
Expanding the factorial term:
$ 3! = 3 \times 2 \times 1 = 6 $
Substituting this back into the series:
$ \sin(z) = z - \frac{z^3}{6} + \frac{z^5}{120} - \dots $
Identifying the Coefficient
In the Taylor series expansion above, we need to find the coefficient of the $z^3$ term. Observing the series:
$ \sin(z) = 1 \cdot z + \left(-\frac{1}{6}\right) \cdot z^3 + \frac{1}{120} \cdot z^5 - \dots $
The coefficient of the $z^3$ term is $C = -\frac{1}{6}$.
Calculating the Reciprocal
The question asks for the reciprocal of this coefficient.
Reciprocal = $ \frac{1}{C} = \frac{1}{-\frac{1}{6}} $
Calculating the reciprocal:
$ \frac{1}{-\frac{1}{6}} = -6 $
The reciprocal of the coefficient of $z^3$ is -6.
Important Questions from Series
- In the sequence 6, 9, 14, $x$, 30, 41, a possible value of $x$ is
Let $a_0 = 0$ and define $a_n = \frac{1}{2}(1 + a_{n-1})$ for all positive integers $n \ge 1$.
The least value of $n$ for which $|1 - a_n| < \frac{1}{2^{10}}$ is __________.
(Answer in integer)
- Let $a_1 = 1$ and $a_n = a_{n-1} + 4$, $n \ge 2$. Then,
$\lim_{n\to\infty} \left[\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \dots + \frac{1}{a_{n-1}a_n}\right]$
is equal to ________ - Let $S(x) = a_0 + \sum_{n=1}^\infty(a_n \cos (n x) + b_n \sin (n x))$ be the Fourier series of the$2 \pi$ periodic function defined by $f(x) = x^2 + 4 \sin (x) \cos(x)$, $-\pi \le x \le \pi$. Then
$|\sum_{n=0}^\infty a_n - \sum_{n=1}^\infty b_n|$
is equal to ________ Let $S_n = \sum_{k=1}^n \frac{1}{k}$ and $I_n = \int_1^n \frac{x - [x]}{x^2} dx$. Then, $S_{10} + I_{10}$ is equal to
