Determine whether the series is convergent or divergent:
a) ∑ from n = 1 to ∞[ 1/nlnn]
b) ∑ from n = 1 to ∞ [1/sqrt(n^2 + 1)]
c) ∑ from n = 1 to ∞ [cos^2n/ n^2 + 1]
d) ∑ from n = 1 to ∞ [2 + (-1)^n/nsqrt(n)]
a) use the sum of the first 10 terms to estimate the sum of the series ∑ from n = 1 to ∞ [1/n^2]. How good is this estimate
b) Improve this estimate using (4) with n = 10
c) find a value of n that will ensure that the error in the approximation S ~ Sn is less than 0.001
a) Answer: divergent
Let f(x)=1/(xlnx), the integration of f(x) from 2 to N is
ln(lnN) -ln(ln2) because the antiderivative of f(x) is ln(lnx).
Now if we consider the upper Riemann sum with respect to the partition
2<3<4<...<n-1<n, then it is exactly ∑ from n = 2 to N[ 1/nlnn].
Let N->oo, then ∑ from n = 2 to ∞[ 1/nlnn] > ln(lnN)-ln(ln2)->oo.
Thus the series ...
Whether series are convergent or divergent is investigated.