Show that C_{0} –2^{2}C_{1}
+ 3^{2}C_{2} - . . . +(–1)^{n}(n+1)^{2}C_{n}
= 0, n> 2.

We have

C_{0} + C_{1}x
+ C_{2}x^{2 }+ . . . + C_{n}x^{n} = (1+x)^{n}
. . . (1).

Multiplying (1) by x

C_{0}x + C_{1}x^{2}+C_{2}x^{3}
+. . . +C_{n}x^{n+1} = x(1+x)^{n} . . .
(2).

Differentiating (2) w.r.t. x, we have .

C_{0} + 2c_{1}x +3C_{2}x^{2}
+ . . . +(n+1) C_{n}x^{n} = (1+x)^{n} + nx(1+x)^{n–1}
.

= [(n+1)x+1] (1+x)^{n–1}
. . . (3).

Multiplying (3) by x, we get

C_{0}x + 2C_{1}x^{2}
+ 3C_{2}x^{3} + . . . + (n+1) C_{n}x^{n–1} =
[(n+1)x^{2}+x](1+x)^{n-1 }. . . (4).

Differentiating (4) w.r.t x , we get .

C_{0} + 2^{2}
C_{1}x + 3^{2} C_{2}x^{2} + . . . + (n+1)^{2}
C_{n}x^{n}.

= [2(n+1)x+1)] (1+x)^{n–1}
+ (n–1) [(n+1)x^{2} + x] ´
(1+x)^{n–2} . . . (5).

Putting x = –1 in (5) , we get

C_{0}–2^{2}C_{1}
+ 3^{2}C_{2} - . . . +(–1)^{n} (n+1)^{2} C_{n}
= 0.

Prove that C_{0}C_{r} + C_{1}C_{r+1}
+ C_{2}C_{r+2} + . . . + C_{n–r} C_{n} =

We have,

C_{0} + C_{1}x
+ C_{2}x^{2} + . . . + C_{n}x^{n }= (1+x)^{n}
. . . (1).

Also, C_{0}x^{n}
+ C_{1}x^{n–1}+ C_{2}x^{n–2} + . . . + C_{n}
= (x+1)^{n} . . . (2).

Multiplying (1) and (2), we get

(C_{0}+C_{1}x+C_{2}x^{2}
+…+ C_{n}x^{n}) (C_{0}x^{n}+ C_{1}x^{n–1}
+C_{2}x^{n–2} + …+C_{n})^{ }

= (1+x) ^{2n }.
. . (3).

Equating coefficinet
of x^{n–r} from both sides of (3), we get

C_{0}C_{r}
+ C_{1}C_{r+1} + C_{2}C_{r+2 + ------------+ }C_{n–r}C_{n}
= ^{2n} C_{n–r = }

Find the sum of the series

The given series

Now, =

Similarly, etc.

Hence, the gives series =

= =

Let k and n ∈ I^{+} and S_{k} = 1^{k} + 2^{k}
+. . . . + n^{k}, show that , ^{m+1}C_{1}S_{1}
+^{ m+2}C_{2}S_{2}+^{ }. . . . + ^{m+1}C_{m}S_{m}
= (n+1)^{m+1} – (n+1) . .

We have

&⇒

Putting p = 1, 2, 3,…,n and adding, we get

&⇒

Show that .

We have, for r ≥ 0,

= =

Thus, =

= [put r+2 = s]

=

=

= =

But 2n+3 + (-1)^{n}
=

Thus,