Calculus: Analytic Geometry and Calculus, with Vectors

644 Series

involving binomial coefficients. For example, putting s = 3 gives

n

k3 = n3 *[n4(1) + 4n3(-C + 6n2(+) + 4n(0)],

k=1

so

Putting s = 4 gives

so

n(n+1)(2n+1)(3t2+3n-1)

30

Some people spend huge amounts of time working out these formulas by other

methods.

3 Show that setting p = 1, q = n, f(x) = x-1, m = 3, and C1 = y in (12.665)

gives the formula

(1)

4

k3 =n2(n + 1)2

k1^4

n

0 = n4 + 3[ns + 5n4(- ) + 10n3(g) + 10n2(0) +

k=1

V -n(6n4 + 15n3 + iOn2 - 1)

k=1^30

Ik=logn+y+2n-12n2+6

k l

where y is the Euler constant. Show that setting p = 1, q = 10, f(x) = z 1,

and C1 = y in (12.666) gives the formula

B,-J10

X I

(2) y kIlk - log 10- + 4 10 m+1B,, x)dx.

7'2 1

Remark: Knowing that

log 10 = 2.30258 50929 94046

it is not difficult to push a pencil through the calculations by which (2) is used

with m = 10 to obtain the first 10 decimal places in

y = 0.57721 56649 01532 86061

When y is known, (1) enables us to make very close estimates of the sum of

10 or more terms of the harmonic series.

4 Supposing that z > -1, show that putting p = 1, q = n, and f(x)

log (z + x) in (12.663) gives, when m z 2,

n

(1) E log(z+k)=(z+n+y)log(z+n)-(z+-1T)log(z+1)-n+1

k-1

+ m

B,

L

1 - 1 +

I

n (m - 1)!Bm(x)dx.

,1-z (9 - 1)1 (z - (z 1),-1^1 (z + x)"