Analysis 1 (Convergent/Divergent) Ratio/Comparison Test?

Question

Use either the comparison test or ratio test to determine which of the following series converge or diverge

k=1∑∞ (3k^2)-3 / (k^5)+1

k=1∑∞ (k^3/4^k)

k=1∑∞ (1/log(k+1))

k=1∑∞ (1/k + log(k))

Answer 1

Let Sum() represent summation 1 to inf. and lim means lim k-->inf
1) Limit Comparison test
Sum(3/k^3) converges because of p-test
3/k^3> (3k^2-3) / ((k^5)+1) because 3k^5-3k^3 < 3k^5+3
Since Sum(3/k^3) converges Sum((3k^2-3) / ((k^5)+1) ) also convegres

2) Ratio Test
lim |(k+1)^3/ (4*4^k) * (4^k/k^3)| = lim (k+1)^3/(4k^3) = 1/4 <1
Since it's <1, Sum(k^3/4^k) converges

3) Comparison test
Sum(1/(k+1)) diverges because of harmonic test
1/log(k+1) > 1/(k+1) because k+1 > log(k+1)
Therefore Sum(1/log(k+1)) diverges because Sum(1/(k+1)) diverges

4)Comparison test
Sum (1/(2k)) diverges because of p-test
1/(k + log(k)) >1/(2k) because k> log(k)
Therefore Sum( 1/(k + log(k))) diverges because Sum (1/(2k)) diverges

Answer 2

compare with 1/k^3 convergent
Ratio test a_n+1/a_n= (k+1)^3/k^3 *4^k/4^(k+1) ==> 1/4<1 convergent
compare with 1/n 1/log(k+1) > 1/(n+1) divergent
compare with 1/k = 1+log/k ==> divergent
I supposed the denominator is( k+ log k)
if not ,lim ak is not 0 so it is also divergent

Answer 3

India had an intensive to suitable tournament after Australia's first innings batting,i presumed it change into sturdy first innings from Aussies yet India scored so quickly owing to Dhawan and co and Australia's 2d innings capitulation that got here about so straight away on the perfect day can not be defined even by ability of Ian Chappell.Batting for India clicked so nicely in this sequence,that they had an intensive to suitable tournament.Australia could've drawn this tournament yet they ended up dropping on the perfect day owing to three undesirable batting in 2d innings.