Advanced book on Mathematics Olympiad

328 Methods of Proof

(

1 +

1

n

)n

< 3 ,

while the inequality on the right can be reduced to

(

1 +

1

n

)n

> 2.

These are both true for alln≥1 because the sequence( 1 +^1 n)nis increasing and converges

toe, which is less than 3. Hence the conclusion.

16.The left-hand side grows withn, while the right-hand side stays constant, so apparently

a proof by induction would fail. It works, however, if we sharpen the inequality to

1 +

1

23

+

1

33

+···+

1

n^3

<

3

2

−

1

n

,n≥ 2.

As such, the casesn=1 andn=2 need to be treated separately, and they are easy to
check.
The base case is forn=3: 1+ 213 + 313 < 1 +^18 + 271 <^32 −^13. For the inductive
step, note that from

1 +

1

23

+

1

33

+···+

1

n^3

<

3

2

−

1

n

, for somen≥ 3 ,

we obtain

1 +

1

23

+

1

33

+···+

1

n^3

+

1

(n+ 1 )^3

<

3

2

−

1

n

+

1

(n+ 1 )^3

.

All we need to check is

3

2

−

1

n

+

1

(n+ 1 )^3

<

3

2

−

1

(n+ 1 )

,

which is equivalent to

1

(n+ 1 )^3

<

1

n

−

1

(n+ 1 )

,

or

1

(n+ 1 )^3

<

1

n(n+ 1 )

.

This is true, completing the inductive step. This proves the inequality.