Show that the sequence of general term a1=a, an+1= is convergent and find its limit

6. If 0 < a < b < 1, show that the series a + b + a² + b² + a³ + b³ + ... is convergent.

Study the convergence of the series (1/3)2 + (1*4/3*6)2 + (1*4*7/3*6*9)2 +...

A well-known approxiamation for r is 22/7=3+1/7. As part of an effort to approximate r, write 1/7 as the sum of a GP.

4. While setting up a game of bowling, someone decided to increase the same, but non-standard, number of pins in each successive row. We know they placed 4 pins in the first row, 12 pins in the last row, and used in total 40 pins. However we cannot see from the end of the alley how many rows were placed, nor many pins were added between rows. Determine those two quantities.

5. A geometric sequence has a third term of 20 and seventh term of 324/125. Find the two possible common ratios, and the initial term.

(3) Determine all values of p and q for which the following series converges: \sum_{k=2}^{\infty} \frac{1}{k^{a}(\ln k)^{p}}

Denote the Fourier series of f(x)=\left\{\begin{array}{ll} -x, & -\pi \leq x<0 \\ x, & 0 \leq x<\pi \end{array}\right. by F(x), where F(x) has a period 27. Show that F(x)=\frac{\pi}{2}-\frac{4}{\pi} \sum_{m=1}^{\infty} \frac{\cos (2 m-1) x}{(2 m-1)^{2}} \text { For }-2 \pi<x<2 \pi \text { siketch the graph of } F(x) Deduce that 1+\frac{1}{3^{2}}+\frac{1}{5^{2}}+\cdots=\frac{\pi^{2}}{8}

\text { 5. Evaluate the sum } S=\sum_{n=1}^{\infty} \frac{-2}{n^{2}+3 n+2}

6. [3 mark] State whether or not the following functions have a well-defined inverse. If the inverse is well-defined, define it. If it is not well-defined, provide justification. \begin{array}{l} \text { (a) } f: z \rightarrow z_{1} f(x)=7 x-7 \\ \text { (a) } f: \mathbf{R} \rightarrow \mathbf{R} \text { . } f(x)=7 x-7 \end{array} (c) A = {a, b, c, d,e}. f: P(A) → {0,1,2, 3, 4,5}. f(2) = |2|. It maps a set to the number of elements it contains.