QUICK CALC QUESITON Show that the series from 1 to infinity (summation with infi
ID: 2846052 • Letter: Q
Question
QUICK CALC QUESITON
Show that the series from 1 to infinity (summation with infinity on top and n = 1 on bottom )
of
1 / n(ln n)^4 converges by using the INTEGRAL TEST. (that is n times the ln of n to the fourth power)
MUST ANSWER THESE QUESTIONS :
A. is the function positive ?
B. Is function continuous on [2,infinity)
C. Show that f(x) is decreasing on [2,infinity)
D. Show that the series converges by applying the integral test, i.e. showing that the integral from 2 to infinity of f(x)dx converges.
Explanation / Answer
A)
yes the function is positive because ln(n)^4 >= 0 and n > 1 -> 1/n.ln(n)^4 > 0
B)
Yes because the denominator of the function is only zero at n = 0 and n = 1 which are not in [2 , +infinity)
C)
f(x) = 1/x.ln(x)^4 -> f'(x) < 0 -> f(x) is decreasing
D)
int 1/x.ln(x)^4 dx = -1/3 (ln(x))^(-3) (2 < x < infinity) = 0 - (-1/3 (ln(2))^(-3)) = ln(2)^(-3)/3