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 .

Answer 2

Hence we cannot use the integral test.

Integral Test:

In the mathematical domain, the Integral test for convergence is a technique that is often applied for the purpose of testing an infinite series of non-negative terms for convergence.

Given function is,

$f(x) = 5 cos(\pi x)$

This is an alternating series.

So we need to do the alternating series test, not the integral test.

The integral test is for the positive series, while this series alternate between positive and negative.

Hence we cannot use the integral test.

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