Raabe’s criterion states: Suppose an> 0 for all n=1,2, … Then

3. Find the limits of the following sequences, or show that the limit does not exist. \text { (a) }\left\{a_{k}\right\}_{k=1}^{\infty} \text { where } a_{k}=\frac{b_{1}}{b_{k-1}} \text { and } b_{k}=\frac{(2 k) !}{h !^{2}} \text { (c) }\left\{a_{n}\right\}_{n=1}^{\infty} a_{n}=\frac{1}{2}\left(a_{n-1}-\frac{4}{a_{n-1}}\right) a_{n}=\frac{n^{3}-n^{2}+n-1}{\sqrt{n^{2}+n+1}-\sqrt{n}} \text { (d) }\left\{a_{n}\right\}_{n=1}^{\infty} \text { where } where a1 = 1 and an is defined by the recurrence \text { (b) }\left\{a_{k}\right\}_{k=1}^{\infty} where a1 = 2 and an is defined by the recurrence a_{n}=\frac{1}{3}\left(2 a_{n-1}+\frac{2}{a_{n-1}^{2}}\right)

2. Do the following series converge or diverge? Write one sentence to explain your answer,and specify which test you used [3 marks each]. \text { (b) } \sum_{n=0}^{\infty} \frac{1}{n \sqrt{n}+3} \text { (c) } \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}+n} \text { (d) } \sum_{n=3}^{\infty} \frac{(\ln n)^{3}}{n} \sum_{n=1}^{\infty} \frac{(-1)^{n}-1}{n}

Determine the Taylor series of the given function. Find the radius of convergence. f(x)=\cos \left(2 x^{3}\right)

Bob makes his first $1,600 deposit into an IRA earning 6.7% compounded annually on his 24th birthday and his last $1,600 deposit on his Next question qual

(4) Suppose series \sum_{k=1}^{\infty} a_{k}, a_{k} \geq 0 is convergent. For each of the following series, either show that it is convergent, or give an example that is divergent. \text { (a) } \sum_{k=1}^{\infty} \frac{a_{k}^{2}}{a_{k}+\frac{1}{k}} \text { (b) } \sum_{k=1}^{\infty} k a_{k}^{2} \text { (c) } \sum_{k=1}^{\infty} \frac{\left(1+\frac{1}{k}+a_{k}\right)^{4}-1}{k+a_{k}} \text { (d) } \sum_{k=1}^{\infty} \sqrt{\frac{a_{k}}{k}} \text { . }

\text { 2. Let }\left\{a_{i}\right\}_{i=0}^{\infty} a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{4}{a_{n}}\right) be the sequence starting at ao = 4, with recurrence relation (a) Prove whether the sequence is increasing, decreasing,or neither. (b) Prove whether or not the sequence is bounded. (c) If possible, use the above to find the limit of the sequence.

If a person borrows $2,400 and repays the loan by paying $200 per month to reduce the loan and 1% of the unpaid balance each month for the use of the money, what is the total cost of the loan over 12 months?

(2) Test each of the following series for convergence: \text { (a) } \sum_{k=1}^{\infty} \frac{\sqrt{k+1}-\sqrt{k}}{k} \text { (b) } \sum_{k=1}^{\infty}(\sqrt[b]{k}-1)^{k} \text { (c) } \sum_{k=1}^{\infty} \sin \left(\frac{\ln k}{k^{2}}\right) \text { (d) } \sum_{k=1}^{\infty} \frac{1}{k} \ln \left(1+\frac{1}{k}\right)

: Find the arc length of the given polar curve r=\csc \theta \quad \frac{\pi}{3} \leq \theta \leq \frac{2 \pi}{3}