Calculus: Art Of Problem Solving (2), (3), (4)

• Aug 7th 2008, 08:34 AM
NonCommAlg
Calculus: Art Of Problem Solving (2), (3), (4)
Hint for Problem 1: the answer is $2n.$ Here are 3 more problems:

$\star \star \star$ Problem 2: Evaluate $\lim_{n\to\infty} \left[\prod_{k=1}^n \binom{n+k}{n}\right]^{n^{-2}}.$

$\star \star$ Problem 3: Evaluate $\lim_{n\to\infty} \frac{1}{n^k} \int_1^b \prod_{j=1}^k \ln(1 + a_jx^n) \ dx$, where $k, \ a_j > 0, \ b > 1$ are given constants.

$\star \star$ Problem 4: For any real number $x>0$ define $f(x)$ to be the number of elements of the set $\{(m,n) \in \mathbb{Z} \times \mathbb{Z}: \ m^2 + n^2 < x \}.$

Find $\lim_{x\to\infty}\frac{f(x)}{x}.$
• Aug 7th 2008, 09:43 AM
CaptainBlack
Quote:

Originally Posted by NonCommAlg

$\star \star$ Problem 4: For any real number $x>0$ define $f(x)$ to be the number of elements of the set $\{(m,n) \in \mathbb{Z} \times \mathbb{Z}: \ m^2 + n^2 < x \}.$

Find $\lim_{x\to\infty}\frac{f(x)}{x}.$

This hardly needs a thought:

$\lim_{x\to\infty}\frac{f(x)}{x}=\pi$

RonL
• Aug 7th 2008, 10:24 AM
NonCommAlg
Quote:

Originally Posted by CaptainBlack
This hardly needs a thought:

$\lim_{x\to\infty}\frac{f(x)}{x}=\pi$

RonL

the answer is correct but the problem is not trivial at all!
• Aug 7th 2008, 11:56 AM
particlejohn
Which book are you using? Is the the AoPs volume 1 and volume 2?
• Aug 7th 2008, 12:07 PM
CaptainBlack
Quote:

Originally Posted by NonCommAlg
the answer is correct but the problem is not trivial at all!

But it is, it is equivalent to the definition of area as the limit of the sum of the area of the squares of a regular rectangular partition of the circle.

RonL