Advanced High-School Mathematics

SECTION 5.2 Numerical Series 269

5. Prove that the limit limn→∞

Ñ n

∑

k=1

1

k

−lnn

é

exists; its limit is called

Euler’s constant^8 and is denotedγ(≈ 0 .577). To prove this just

draw a picture, observing that the sequence^12 <

∑n

k=1

1

k

−lnn < 1

for allnand that the sequencean=

∑n

k=1

1

k

−lnnis andecreasing

sequence.^9

5.2.2 Tests for convergence of non-negative term series

In this subsection we’ll gather together a few handy tests for conver-
gence (or divergence). They are pretty intuitive, but still require prac-
tice.

The Limit Comparison Test

(i) Let

∑∞

n=0

an be aconvergentseries of positve terms and let

∑∞

n=0

bn

be a second series of positive terms. If for someR, 0 ≤R <∞

nlim→∞

bn

an

=R,

then

∑∞

n=0

bn also converges. (This is reasonable as it says that

(^8) or sometimes theEuler-Mascheroniconstant
(^9) Drawing a picture shows that
1 + 21 +^13 +···+n−^11 −lnn≥^14 +^12
Å 1
2 −
1
3
ã
+^12
Å 1
3 −
1
4
ã
+···+^12
Å 1
n− 1 −
1
n
ã
=^12 − 21 n.
Therefore,
1 +^12 +^13 +···+^1 n−lnn≥^12 + 21 n>^12.
Next, that the sequence is decreasing follows from the simple observation that for alln >0,
1
n+ 1<ln
Ån+ 1
n
ã
.
Finally, I’d like to mention in passing that, unlike the famous mathematical constantsπande
(which are not only irrational but actually transcendental), it is not even known whetherγis rational
or irrational.