Question

Consider the following function. f(x) = 5 cos(πx) x What conclusions can be made about the series [infinity] 5 cos(πn) n n = 1 and the Integral Test? The Integral Test can be used to determine whether the series is convergent since the function is positive and decreasing on [1, [infinity]). The Integral Test can be used to determine whether the series is convergent since the function is not positive and not decreasing on [1, [infinity]). The Integral Test can be used to determine whether the series is convergent since it does not matter if the function is positive or decreasing on [1, [infinity]). The Integral Test cannot be used to determine whether the series is convergent since the function is not positive and not decreasing on [1, [infinity]). There is not enough information to determine whether or not the Integral Test can be used or not.

Answer 1

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 $\frac{(-1)^n}{n}$.

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 ,

$\lim_{n \to \infty} \int\limits^({1/x} \, dx$    =infinity ,with limits as (1 to n+1)

$\int\limits^a_b {(1/x)} \, dx$    =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 .