8.2 Nonnegative Series 47500 1- Prove that ; n(ln n) [ln(ln n) JP converges if and only if p > l.
L
oo Inn- Prove that - converges if and only if p > l.
nP
n=2
.. J2ln4 v'5ln 7 VSln 10 VUln 13 - Determme whether the senes ---+---+ + +· ..
1·2 2·3 3. 4 4.5
converges or diverges. [Hint: Use Exercise 33 .] - Prove that both the ratio test and the root test are inconclusive for the
p-senes,. '"""' L; kP^1. - Prove Theorem 8.2.11.
- Prove Theorem 8.2.13. [Use Theorems 2.9.7 and 8.2.10.J
- Prove Theorem 8.2.15. [See proof of Exercise 39.J
- Prove Theorem 8 .2. 16 [Use Theorems 2.9.7 and 8.2.14.J
- Complete Part 2 of the proof of Theorem 8.2.18.
- Euler's constant ( ')' = }~1!, (1 + ~ + ~ + · · · + ~) -ln n): Although
the harmonic series 2::: t diverges, there is an interesting relationship be-
tween its partial sums and ln n. In fact, we shall show that as n --+ oo their
difference converges to a constant, denoted ')', called Euler's constant.
Define the sequence bn} by
'Yn = (1 + ~ + ~ + · · · + ~) -ln n.
Using techniques used in the proof of the integral test (8.2.3) show that
'Yn > 0, and hence bn} is bounded below. To see that bn} is monotone
decreasing, start by showing that'Yn+l - 'Yn -- n+l^1 - 1 ( n n + + 1)^1 nn -_ n+l^1 - ln(n+l)-lnn (n+l)-n ,
and then apply the mean value theorem. Conclude that 'Y = lim 'Yn
n-+oo
exists. b is approximately 0.557215665, to nine decimal places. It is not
known whether this number is rational or irrational.]- Prove Corollary 8.2.21.
~ 1. 3. 5. 7 ..... (2k - 1)
- Use Raabe's test to prove that ~ 2. 4.
6
.
8
..... ( 2 k) diverges.
k=l