2. (a) Show that the sequence (an)n∈N, where an = (-1)^n / 5^n is a Cauchy sequence. (b) Show that the function f defined on [0, 1] by f(x) = x, when x = 1/n for some n ∈ N; 0, els... 2. (a) Show that the sequence (an)n∈N, where an = (-1)^n / 5^n is a Cauchy sequence. (b) Show that the function f defined on [0, 1] by f(x) = x, when x = 1/n for some n ∈ N; 0, elsewhere is Riemann integrable. (c) State the Fundamental Theorem of Calculus. Use it to evaluate ∫0^1 3x dx. 3. (a) Prove that n^3 + 2n is divisible by 3 for all n ≥ 1, by mathematical induction. (b) Let f be a function defined on R as: f(x) = 1, when x is rational; 0, when x is irrational. Prove that f is discontinuous at each x ∈ R. (c) Define pointwise convergence for a sequence of functions. Check whether the sequence (fn)n∈N, where fn(x) = cos(2nx^3)/n^3, x ∈ [0, 1] is pointwise convergent or not. Read more

Steps to Solve

1. Show that the sequence is Cauchy To prove that the sequence ( a_n = \frac{(-1)^n}{5^n} ) is a Cauchy sequence, we need to show that for every ( \epsilon > 0 ), there exists ( N ) such that for all ( m, n \geq N ), the inequality ( |a_n - a_m| < \epsilon ) holds.

2. Evaluate ( |a_n - a_m| ) Calculate the difference: [ |a_n - a_m| = \left| \frac{(-1)^n}{5^n} - \frac{(-1)^m}{5^m} \right| ] This can be rewritten using a common denominator: [ = \left| \frac{(-1)^n 5^m - (-1)^m 5^n}{5^{n+m}} \right| ]

3. Simplify the expression Factor the numerator: [ = \frac{1}{5^{n+m}} \left| (-1)^n 5^m - (-1)^m 5^n \right| ] Estimate the absolute value. Notice that ( |(-1)^n 5^m - (-1)^m 5^n| \leq 5^m + 5^n \leq 2 \cdot 5^{\max(n,m)} ).

4. Find ( N ) based on ( 5^n ) We want: [ \frac{2 \cdot 5^{\max(n,m)}}{5^{n+m}} < \epsilon ] This leads to: [ 2 < \epsilon \cdot \frac{5^{n+m}}{5^{\max(n,m)}} ] Choose ( N ) such that ( \frac{5^{n+m}}{5^{\max(n,m)}} ) becomes large enough to satisfy the above condition as ( n ) and ( m ) increase.

5. Conclusion Since the sequence converges to 0, it is a Cauchy sequence.

6. Show Riemann Integrability To determine the Riemann integrability of the function ( f(x) ), note that it only takes the value ( x ) at countably infinite points (i.e., ( \frac{1}{n} )), and ( 0 ) elsewhere. Since the set of discontinuities is countable, the function is Riemann integrable.

7. Use the Fundamental Theorem of Calculus To evaluate the integral ( \int_0^1 3x , dx ): First, find the antiderivative: [ F(x) = \frac{3x^2}{2} ] Apply the limits: [ \int_0^1 3x , dx = F(1) - F(0) = \frac{3(1)^2}{2} - 0 = \frac{3}{2} ]