BCA Semester 1 – Mathematics I – Unit 3: Sequence and Series

Find the Maclaurin series of the function $f(x)=sin⁡x.$

If H is the harmonic mean between a and b, prove that:

A class consists of boys whose ages are in A.P. (common difference = $4$ months).
The youngest boy is $8$ years old, and the total age of the class is $168$ years.
Find the number of boys.

a) Three numbers in A.P. sum to $15$. If $1, 3, 9$ are added to them respectively, they form a G.P. Find the original numbers.
b) Find the sum to $n$ terms of the series

If $A$ is the A.M. and $H$ is the H.M. between two numbers $a$ and $b$, show that:

a) Find the Taylor series expansion of $f(x) = x^3 - 2x + 4$ at $a = 2$.

b) In how many ways can the letters of the word $\text{'CALCULUS'}$ be arranged so that the two $\text{C'}$s do not come together?

Expand $e^x$ about $x = 0$ using the Maclaurin series.

If $a, b, c, d$ are in G.P., prove that$a^2 - b^2, \; b^2 - c^2, \; c^2 - d^2$ are also in G.P.

a) Find the Maclaurin series of the function: $f(x) = \cos x$
b) Take any matrix of order $3 \times 3$ and express it as a sum of a symmetric and a skew-symmetric matrix.