Determine whether the series is absolutely convergent condit

Determine whether the series is absolutely convergent, conditionally convergent or divergent: sigma_n=1^infinity 1/n^2 cos npi/7

Solution

a convergent series An is said to be absolutely convergent if |An| is also convergent , otherwise it is called a conditionally convergent series.

for given problem An =sum[1/n² * cos(n pi/7)]

we know that -1<= cos x <=1 for any value of x

and for 1/n² infinite series, convergence can be tested using the integral test. If the integral of the function of the sequence is convergent, then the series is convergent. If the integral is divergent, the series is divergent. Integral from 1 to infinity (1/n²) dn = Limit t->infinity Integral 1 to t (1/n²) dn = Lim t->infinity -1/n |1 to t = Lim t->infinity -1 - (-1/t) = -1 The integral is finite, therefore convergent

as both 1/n² converges and cos npi/7 comes in [-1,1] so there product An is also convergent .

now for |An| we have to test (1/|n²| ) and |cos npi/7|

(1/|n²| is convergent for positive intervals as it is same as 1/n² in(0,infinity)

and |cos npi/7| belongs to [0, 1]

|cos npi/7|/n² < 1/n² as 1/n² covergerges so original series |cos npi/7|/n2 converges hence |An| converges

hence An is absolutely convergent