Real Analysis/Section 2 Exercises/Hints

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Real Analysis/Section 2 Exercises Exercises

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These are a list of problems for the Sequences and Series section of the wikibook. The questions listed here will also contain questions related to Taylor Polynomials and Taylor Series, as they contain sequence and series like qualities. Do those problems only if you have some knowledge on derivatives. Otherwise, these following problems should not rely on too many concepts outside of its section.

Unsorted

1. For the following list of sequences determine whether the following sequences converge or diverge directly from the definition of convergence:
   1. the sequence $\textstyle ({\frac {n-1}{n}})_{n=2}^{\infty }$;
   2. the sequence $\textstyle ({\frac {{\sqrt {2n-1}}+2}{\sqrt {n+3}}})_{n=1}^{\infty }$;
   3. the sequence $\textstyle \left({\frac {2n-7}{n^{2}+1}}\right)_{n=1}^{\infty }$;
   4. The recursive sequence defined by the following:

      $x_{0}=2$

      $x_{n}={\sqrt {x_{n-1}}}$

2. Let

   $x_{n}={\begin{cases}{\frac {n-1}{n}}&{\text{if }}n{\text{ is odd;}}\\{\frac {1}{n}}&{\text{if }}n{\text{ is even.}}\end{cases}}$.

   Find lim sup x_n.
3. For the following sequences determine for which real values x does the given sequence converge, and what the sequence converges to:
   1. $\textstyle ({\frac {x}{n}})_{n=1}^{\infty }$;
   2. $\textstyle (x^{n})_{n=1}^{\infty }$;
   3. $\textstyle ({\frac {1}{n^{x}}})$ for an arbitrary real number x;
4. Given any real number c, find a recursively defined sequence that converges to ${\sqrt {c}}$.
5. Given a sequence (x_n), and a natural number k, define a sequence y_n by y_n = x_n+k. Show that (x_n) is convergent if and only if (y_n) is convergent. Show further that when they converge they converge to the same limit.
6. Suppose that the sequences (x_n) and (y_n) converge to a real number a. Show that the sequence (z_n) defined by

   $z_{n}={\begin{cases}x_{2n-1}&{\text{if }}n{\text{ is odd;}}\\y_{2n-3}&{\text{if }}n{\text{ is even.}}\end{cases}}$

   Show that z_n → a.
7. Let (x_n) be a sequence of real numbers and let (y_n) be a sub-sequence. Suppose (y_n) is convergent, show that (x_n) may not necessarily be convergent.
8. Suppose that (x_n) is a convergent sequence that does not converge to 0. Further assume that for all n in N, x_n ≠ 0. Show that there exists δ > 0 so that |x_n| > δ and |lim x_n | > δ.
9. Cesaro Mean convergence: We say a sequence (x_n) converges to x by Cesaro means if the sequence of averages y_n = (x₁ + x₂ + … + x_n)/n converges to x. Suppose (x_n) converges to a real number x, show that (x_n) converges by Cesaro means to x. Give an example to show that a divergent sequence (x_n) may converge by Cesaro means.
10. Find the sequence of Cesaro means for (1, 1, -1, 1, 1, -1...) and determine if the converge. If they converge, find the limit.
11. Consider the recursively defined sequence given by x₁ = 1 and x_n = 1 + 1/x_n. Show that x_n converges and find its limit.
12. In our discussion of telescoping series we showed that a telescoping series converged to a₁ − lim a_N+1 and for this to hold it was not necessary to have that lim a_N+1 = 0. Indeed, it is correct that this is not necessary. On the other hand we later proved that for a convergent series the limit of the terms must be 0. How can both by correct? Explain why, in our set up, we may have a convergent telescoping series such that lim a_N+1 ≠ 0, but it is still true that for every convergent series the limit of the terms is 0.
13. Suppose that c_n ≤ a_n ≤ b_n for all natural numbers n. Show that if both ∑ c_n and ∑ b_n converge, then ∑ a_n converges.

On Sequences

For the questions below, clicking on the box will reveal a general hint and options to select different types of hints. The solution will also be present, but hidden until clicked. The answer to each question is on a separate page, for both technical and psychological reasons.

1. Given the sequence {a_n} such that {a_i} = i/i+1, prove that {a_n} is strictly increasing.

Question Hint

Prove that a_i < a_i+1 for all i, a natural number.

Methodology Hint 1

It is easy to prove that the sequence strictly increasing using mathematical induction.

Methodology Hint 2

During mathematical induction, you may need to multiply out both sides.

Answer

2. Prove that the natural numbers have no upper bound

Question Hint

Consequence of N: Given n, there exists n + 1 > n

Methodology Hint

Prove, using its definition and contradiction, that an upper bound cannot exists for natural numbers

Taylor Polynomials and Series

1. The following questions do not rely on any theorems related to Taylor Polynomials. However, you should know what the definition of a Taylor Polynomial is.
   1. Prove that any Taylor Polynomial is continuous.
   2. Prove that any Taylor Polynomial is infinitely differentiable.

Hints / Answers

1. No Hint.
2. No Hint.
3. No Hint.
4. Use Newton's method to approximate the zeros of the function x² − c.
5. No Hint.
6. No Hint.
7. No Hint.
8. No Hint.
9. No Hint.
10. No Hint.
11. To show that the sequence converges show that the sequence is bounded and contains two monotone subsequences, each of which converge by the monotone convergence theorem, and whose difference converges to 0. To find the limit, take limits of both sides of the recursion relation to find an equation the limit must satisfy.
12. No Hint.
13. Look carefully at how we defined a telescoping sum.
14. Try to relate this back to the usual comparison test.