Question

use the direct comparison test to determine whether the series converges or diverges

4. Use the direct comparison test to determine whether the series converges or diverges. (8 points) Š n 2n3 + 1

Answer 1

Here we will use two test

1) Comparison test

Suppose that we have two series
an
and
bn.
with a_n,b_n≥0a for all n and a_n≤b_n for all n. Then,

a) If
bn.
is convergent then so is
an

b) If
an
is divergent then so is
bn.

2) integral test for convergent /divergent

to check the series will be convergent or divergent

Σ n=

we will take a function f(n) such that f(n) = a_nthen will replace n by x to get function f(x)

If f(x)  is a continuous, positive and decreasing function on the interval [k,infinity ) ,

then  
Σ n=
will be convergent if  
f(x)dc
  is convergent ( i.e. integral have a finite value )

and  
Σ n=
  will be divergent if  
f(x)dc
   is divergent ( i.e. integral have a infinite value )

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