###### A Mathematical Orchard: Problems and Solutions

208 SOLVED PROBLEMS

Question: P.177

## Let c =∑n=1^∞ 1/n(2^n − 1) = 1 +1/6+1/21+1/60+ · · ·. Show that e^c =2/1·4/3·8/7·16/15· · · · . ...

Let $P_k$ be the partial product [lat...
Question: P.176

## Let n ≥ 3 be a positive integer. Begin with a circle with n marks about it. Starting at a given point on the circle, move clockwise, skipping over the next two marks and placing a new mark; the circle now has n + 1 marks. Repeat the procedure beginning at the new mark. Must a mark eventually appear ...

To show why, suppose there is a pair m_1, m...
Question: P.175

## Let f be a continuous function on [0, 1], which is bounded below by 1, but is not identically 1. Let R be the region in the plane given by 0 ≤ x ≤ 1, 1 ≤ y ≤ f(x). Let R1 = {(x, y) ∈ R|y ≤ y} and R2 = {(x, y) ∈ R|y ≥ y}, where y is the y-coordinate of the centroid of R. Can the volume obtained by ...

Solution 1. To find an example, we let R be the tr...
Question: P.174

## Let x0 be a rational number, and let (xn)n≥0 be the sequence defined recursively by xn+1 =|2xn³/3xn² − 4|. Prove that this sequence converges, and find its limit as a function of  x0. ...

\lim_{n\to\infty}x_{n}={\left\{\begin{array...
Question: P.173

## Find lim n→∞ ∫0^∞ n cos( √^4 x/n²)/1 + n²x²dx. Idea. First, we note that we cannot switch the limit and the integral, for if we could, we could start by showing that lim n→∞ n cos(√^4 x/n²)/1 + n²x² = 0 for any nonzero x, and then conclude that the answer would be 0. This answer is incorrect, and a ...

With the reasoning just presented, we see that it ...
Question: P.172

## Let ABCD be a parallelogram in the plane. Describe and sketch the set of all points P in the plane for which there is an ellipse with the property that the points A, B, C, D, and P all lie on the ellipse. ...

Solution 1. To show why this description is correc...
Question: P.171

## If an insect starts at a random point inside a circular plate of radius R and crawls in a straight line in a random direction until it reaches the edge of the plate, what will be the average distance it travels to the edge? ...

The average distance traveled to the edge of the p...
Question: P.170

## Suppose we start with a Pythagorean triple (a, b, c) of positive integers, that is, positive integers a, b, c such that a² + b² = c² and which can therefore be used as the side lengths of a right triangle. Show that it is not possible to have another Pythagorean triple (b, c, d) with the same ...

Suppose we did have positive integers a, b, c, d s...
Question: P.169