Questions: Q.3/ Determine whether the series convergent or divergent, if it is convergent, find its sum for: (5 M) A. sum from n=1 to infinity of (7^(n+1))/(2^n) B. sum from n=2 to infinity of 2/((n-1)(n+1)) Q.4/ Determine whether the series convergent or divergent: (5 M) A. sum from n=1 to infinity of n^0.1/n^2 B. sum from n=1 to infinity of ((-1)^(n-1))/n

Q.3/ Determine whether the series convergent or divergent, if it is convergent, find its sum for:
(5 M)
A. sum from n=1 to infinity of (7^(n+1))/(2^n)
B. sum from n=2 to infinity of 2/((n-1)(n+1))
Q.4/ Determine whether the series convergent or divergent:
(5 M)
A. sum from n=1 to infinity of n^0.1/n^2
B. sum from n=1 to infinity of ((-1)^(n-1))/n

Transcript text: Q.3/ Determine whether the series convergent or divergent, if it is convergent, find its sum for: (5 M) A. $\sum_{n=1}^{\infty} \frac{7^{n+1}}{2^{n}}$ B. $\sum_{n=2}^{\infty} \frac{2}{(n-1)(n+1)}$ Q.4/ Determine whether the series convergent or divergent: (5 M) A. $\sum_{n=1}^{\infty} \frac{n^{0.1}}{n^{2}}$ B. $\quad \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$

Solution

Solution Steps

Step 1: Analyze Series A

For the series $ \sum_{n=1}^{\infty} \frac{7^{n+1}}{2^{n}} $, we identify the first term $ a = 49 $ and the common ratio $ r = 3.5 $. Since $ |r| = 3.5 > 1 $, the series diverges.

Step 2: Analyze Series B

For the series $ \sum_{n=2}^{\infty} \frac{2}{(n-1)(n+1)} $, we find that the sum converges to $ \frac{3}{2} $. This indicates that the series is convergent.