Advanced book on Mathematics Olympiad

3.1 Sequences and Series 123

367.For a nonnegative integerk, defineSk(n)= 1 k+ 2 k+···+nk. Prove that

1 +

∑r−^1

k= 0

(

r

k

)

Sk(n)=(n+ 1 )r.

368.Let

an=

4 n+

√

4 n^2 − 1

√

2 n+ 1 +

√

2 n− 1

, forn≥ 1.

Prove thata 1 +a 2 +···+a 40 is a positive integer.

369.Prove that

∑n

k= 1

(− 1 )k+^1

12 − 22 + 32 −···+(− 1 )k+^1 k^2

=

2 n

n+ 1

.

370.Prove that

(^9999) ∑
n= 1

1

(

√

n+

√

n+ 1 )(^4

√

n+^4

√

n+ 1 )

= 9.

371.Letan=

√

1 +( 1 +^1 n)^2 +

√

1 +( 1 −^1 n)^2 ,n≥1. Prove that

1

a 1

+

1

a 2

+···+

1

a 20

is a positive integer.

372.Evaluate in closed form

∑∞

m= 0

∑∞

n= 0

m!n!

(m+n+ 2 )!

.

373.Letan= 3 n+

√

n^2 −1 andbn= 2 (

√

n^2 −n+

√

n^2 +n),n≥1. Show that

√

a 1 −b 1 +

√

a 2 −b 2 +···+

√

a 49 −b 49 =A+B

√

2 ,

for some integersAandB.

374.Evaluate in closed form

∑n

k= 0

(− 1 )k(n−k)!(n+k)!.