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

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

2. 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

3. 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!

4. Which book are you using? Is the the AoPs volume 1 and volume 2?

5. 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