If Σ an is divergent and Σ bn is convergent, show that Σ (an – bn) is divergent . infinite series problems with solution solved problems in calculus

Solution –  

Assume  \sum (a_n-b_n)    is convergent. Then, \sum a_n=\sum b_n+\sum (a_n-b_n)   is convergent, contrary to hypothesis

---

Q-Find the values of x for which the series Inx + (Inx)^2  + (Inx)^3  + • • • converges and express the sum as a function of x

Solution

lnx+(lnx)^2+(lnx)^3+(lnx)^4+...

The geometric sum formula for infinite terms is given as:

If common ration |r| < 1, S_\infty=\frac{a}{1-r}

If |r| > 1, the series does not converge and it has no sum.

Where

a is the first term of series

r is the common ratio of series

n is the number of terms of series

This is a geometric series with ratio ln x . It converges for |ln x| < 1, -1 < ln x < 1,
1/e < x < e. The sum is \frac{(ln x)}{(1 - ln x)}