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

Given function: \( f(x) = \sin x \) Maclaurin series formula: \( f(x) = f(0) + x f'(0) + \dfrac{x^2}{2!} f''(0) + \dfrac{x^3}{3!} f'''(0) + \cdots \) Derivatives of \( f(x) \): \( f(x) = \sin x \Right

$\textbf{Given:} $ $H$ is the harmonic mean between $a$ and $b$. $\textbf{By definition of harmonic mean:} $ \[ H = \frac{2ab}{a + b} \] $\textbf{To prove:} $ \[ \frac{1}{H - a} + \frac{1}{H - b}

$\textbf{Given:}$ $\text{A class consists of boys whose ages are in A.P. with common difference } d = 4 \text{ months } = \frac{1}{3} \text{ years.}$ $\text{Youngest boy's age } a = 8 \text{ years

$\textbf{Problem a:}$ 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. $\textbf{Step 1:}$ Let the three numbers in A.P. be $a

$\textbf{Given:}$ $A$ is the arithmetic mean (A.M.) and $H$ is the harmonic mean (H.M.) of two numbers $a$ and $b$. $\textbf{Step 1:}$ Recall the formulas: \[ A = \frac{a + b}{2}, \quad H = \fra

$\textbf{Problem a:}$ Find the Taylor series expansion of \[ \quad f(x) = x^3 - 2x + 4 \quad \text{at } a = 2 \] $\textbf{Step 1:}$ Recall the Taylor series formula: \[ \quad f(x) = f(a) + f'(a)(x

$\textbf{Given function:}$ \[ f(x) = e^x \] $\textbf{Step 1:}$ Recall the Maclaurin series formula: \[ f(x) = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 + \cdots =

$\textbf{Given:}$ $a, b, c, d$ are in G.P. $\textbf{Step 1:}$ Let the common ratio of the G.P. be $r$: \[ b = ar, \quad c = ar^2, \quad d = ar^3 \] $\textbf{Step 2:}$ Consider the terms: \[ a^2

$\textbf{Problem a:}$ Find the Maclaurin series of $f(x) = \cos x$. $\textbf{Step 1:}$ Recall Maclaurin series formula: \[ f(x) = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3