- Find lim
_{n→∞}. - Find lim
_{n→∞}. - Find lim
_{n→∞}. - Find lim
_{n→∞}−n. Hint: Multiply and divide by+ n. - Find lim
_{n→∞}∑_{k=1}^{n}. - For < 1, find lim
_{n→∞}∑_{k=0}^{n}r^{k}. Hint: First show ∑_{k=0}^{n}r^{k}=−. Then recall Theorem 3.2.9. - Using the binomial theorem prove that for all n ∈ ℕ,
^{n}≤^{n+1}. Hint: Show first that=. By the binomial theorem,Now consider the term

and note that a similar term occurs in the binomial expansion for^{n+1}except you replace n with n + 1 whereever this occurs. Argue the term got bigger and then note that in the binomial expansion for^{n+1}, there are more terms. - Prove by induction that for all k ≥ 4, 2
^{k}≤ k! - Use the Problems 21 and 7 to verify for all n ∈ ℕ,
^{n}≤ 3. - Prove lim
_{n→∞}^{n}exists and equals a number less than 3. - Using Problem 9, prove n
^{n+1}≥^{n}for all integers, n ≥ 3. - Find lim
_{n→∞}nsinn if it exists. If it does not exist, explain why it does not. - Recall the axiom of completeness states that a set which is bounded above has a
least upper bound and a set which is bounded below has a greatest lower bound.
Show that a monotone decreasing sequence which is bounded below converges to its
greatest lower bound. Hint: Let a denote the greatest lower bound and recall that
because of this, it follows that for all ε > 0 there exist points of in.
- Let A
_{n}= ∑_{k=2}^{n}for n ≥ 2. Show lim_{n→∞}A_{n}exists and find the limit. Hint: Show there exists an upper bound to the A_{n}as follows. - Let H
_{n}= ∑_{k=1}^{n}for n ≥ 2. Show lim_{n→∞}H_{n}exists. Hint: Use the above problem to obtain the existence of an upper bound. - Let I
_{n}=and let J_{n}=. The intervals, I_{n}and J_{n}are open intervals of length 2∕n. Find ∩_{n=1}^{∞}I_{n}and ∩_{n=1}^{∞}J_{n}. Repeat the same problem for I_{n}= (−1∕n,1∕n] and J_{n}= [0,2∕n). - Let be a sequence in. Let A
_{k}≡ supso that λ ≡ limsup_{n→∞}a_{n}= lim_{n→∞}A_{n}, the A_{n}being a decreasing sequence.- Show that in all cases, there exists B
_{n}< A_{n}such that B_{n}is increasing and lim_{n→∞}B_{n}= λ. - Explain why, in all cases there are infinitely many k such that a
_{k}∈. Hint: If for all k ≥ m > n, a_{k}≤ B_{n}, then a_{k}< B_{m}also and so sup≤ B_{m}< A_{m}contrary to the definition of A_{m}. - Explain why there exists a subsequence such that lim
_{k→∞}a_{nk}= λ. - Show that if γ ∈and there is a subsequencesuch that lim
_{k→∞}a_{nk}= γ, then γ ≤ λ.

This shows that limsup

_{n→∞}a_{n}is the largest insuch that some subsequence converges to it. Would it all work if you only assumed thatis not −∞ for infinitely many n? What if a_{n}= −∞ for all n large enough? Isn’t this case fairly easy? The next few problems are similar. - Show that in all cases, there exists B
- Formulate a similar problem which shows that for a sequence of real numbers, liminf
_{n→∞}a_{n}is the smallest number which is obtainable as a limit of a subsequence of the original sequence.

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