Calculus: Analytic Geometry and Calculus, with Vectors

5.5 The Rolle theorem and the mean-value theorem 331

15 Supposing that n is a positive integer and x ; 1, differentiate

• 1

(1) 1+x+x2+x3+. +x' =

xntl

x-1

to obtain

(2) 1 + 2x + 3x2 +. + 0x0-1 =nxn+1 - (n + 12)x0 + 1

(x - 1)

Multiply by x and differentiate again to obtain

(3) 12 + 22x + 32x2 +. + n2xn-1

_n2xn+2 - (202 + 2n - 1)x0+1 + (n + 1)2x0 - x - 1

(x - 1)a

Finally, take limits as x -- 1 to obtain the formula

(4) 12+22+32+ +n2=

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

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Remark: We could multiply (3) by x and differentiate and continue the process

to obtain formulas for sums of cubes and higher powers. The details mushroom

rapidly as we proceed.

16 Another L'Hopital rule is embodied in the following theorem.

(1) Theorem If f and g are differentiable over the infinite interval x Z xo

and if

(2) lim f(x) = oo, lim g(x) = co,

then

(3) Jim f (z) = Jim f ' (x)

'- g(x) x-.- g'(x)

provided the limit on the right exists.

To prove this theorem, suppose that the right member of (3) exists and is L.

Let

(4) O(x) = f(x) - Lg(x).

Then

(5) lim Y(x) = Jimf'(x) - Lg'(x)= 0.

X-.-g'(x) x-.- g'(x)

Let e > 0. Choose x1 such that x1 > xo, g(x*) > 0 when x* '-::t xi, and

(6)

I

'(x*)(x*) <

2

e (x* > xi).

It then follows from the generalized mean-value theorem of Problem 13 that

(7) IOx- O(xi) < e

g(x) - g(xl)

I

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