Series and Sequences

- Series: $\displaystyle\sum{n=2}^{\infty} \frac{2}{n^2-1}$ - Starting index: $n = 2$ - Task: determine convergence/divergence (and sum if convergent). Partial fractions. $$n^2 - 1 = (n-1)(n+1)$$ $$\frac{2}{(n-1)(n+1)} = \frac{A}{n-1} + \frac{B}{n+1}$$ $$2 ...

- Series: $\displaystyle\sum{n=1}^{\infty} an$ where $an = \dfrac{5}{2n^2 + 4n + 3}$ - All terms positive for $n \geq 1$ (numerator and denominator positive). Choose the Limit Comparison Test. For large $n$, the dominant term in the denominator is $2n^2$, s...

- Series: $\displaystyle\sum{n=1}^{\infty}\frac{1}{n^2+1}$ - Associated function: $f(x)=\dfrac{1}{x^2+1}$, tested on $[1,\infty)$ Let $f(x)$ be a function that is continuous, positive, and monotonically decreasing on $[1,\infty)$, and suppose $f(n)=an$ for ...

- Series: $\displaystyle\sum{n=1}^{\infty} \frac{1}{n^2+4}$ - Required: (i) Define integral test [1] (ii) Test convergence/divergence [4] --- Integral Test: Let $f(x)$ be continuous, positive, and decreasing on the interval $[N, \infty)$ for some positive i...

Series: $\displaystyle\sum{n=1}^{\infty} n^2 e^{-n}$ General term: $an = n^2 e^{-n} = \dfrac{n^2}{e^n}$ All terms are positive, so tests for positive-term series (Ratio Test, Root Test) apply. For a positive series, compute $$L = \lim{n \to \infty} \frac{a{...

- Function: $f(x) = \dfrac{1}{x}$ - Center: $a = 2$ $$f(x) = x^{-1}$$ $$f'(x) = -x^{-2}$$ $$f''(x) = 2x^{-3}$$ $$f'''(x) = -6x^{-4}$$ General pattern: $$f^{(n)}(x) = (-1)^n \, n! \, x^{-(n+1)}$$ $$f^{(n)}(2) = (-1)^n \, n! \, 2^{-(n+1)} = \frac{(-1)^n \, n!...

- Function: $f(x) = \dfrac{1}{x}$ - Center of expansion: $a = 2$ Derivatives of $f(x) = x^{-1}$ $$f'(x) = -x^{-2}, \quad f''(x) = 2x^{-3}, \quad f'''(x) = -6x^{-4}$$ General form: $$f^{(n)}(x) = (-1)^n \, n! \, x^{-(n+1)}$$ Evaluate at $a = 2$ $$f(2) = \fra...

- Part 1: $f(x) = x^2 - 2 = 0$, find positive root. - Part 2: $f(x) = e^x$, expand at $x = 0$ (Maclaurin series). - Part 3: $\int1^2 x^2\,dx$, Trapezoidal Rule with $n = 4$. --- (a) Positive Root of $f(x) = x^2 - 2 = 0$ Newton-Raphson Method: $$x{n+1} = xn ...

- Series: $\displaystyle\sum{n=1}^{\infty}\left(\frac{1}{n+1}\right)^n$ - General term: $an = \left(\dfrac{1}{n+1}\right)^n$ - All terms positive, so ordinary convergence = absolute convergence. No data missing. For a positive-term series $\sum an$, define ...

- Series 1: $\displaystyle \sum{n=1}^{\infty} \frac{5}{5n-1}$ - Series 2: $\displaystyle \sum{n=0}^{\infty} \frac{1}{n!}$ --- Step 1 - Divergence (nth term) test. $$\lim{n \to \infty} \frac{5}{5n-1} = 0$$ The nth term tends to $0$, so this test is inconclus...

- Function: $f(x) = \ln x$ - Required: Maclaurin's series expansion (expansion about $x = 0$) The Maclaurin series is a Taylor expansion about $x = 0$: $$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$$ For $f(x) = \ln x$, we r...