Answer:
Hence series can be determine as convergent since the function is positive and decreasing on [1,infinity].
Step-by-step explanation:
Given: the function f(x)=5cos(πx)/x as series from [1,infinity]
To find : Is series positive or decreasing and converges in given range [1,infinity].
Solution:
we have series as : [1,infinity] with function 5cos(πn)/n and 5 being constant
consider the dependent function cos(πn) and 1/n we get ,
by definition cos(πn)=(-1)^n .
hence ,
summation as n[1,infinity] function as .
using alternate series test, series converges as 1/n tends to 0 and decreases ,but
by integral test is not convergent series because :
Sn=1+1/2+1/3+1/4+.........+1/n > integral with limits (1 to n+1) with function (1/x)dx=ln(x) with (1 to n+1) .
hence =ln(n+1)
as n tends to infinity n+1 will be tending infinity.
It is harmonic series ,
=infinity ,with limits as (1 to n+1)
=infinity. with limits as (1 to n+1).
hence we can prove that series convergent or divergent with improper integral .It is called as integral test .