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Concept Quizzes

Analysis Warmup
Sequences and Convergence Warmup
Monotonic Sequences
Subsequence Convergence
Cauchy Sequences
Deep Dive: Pointwise and Uniform Convergence 1
Deep Dive: Pointwise and Uniform Convergence 2

Subsequence Convergence

Which of the following sequences are subsequences of the sequence $a_n=n^2$ ? $\text{I. } a_n=(2n)^2\quad\text{II. } a_n=2n^2.$

Neither I nor II II only I and II I only

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by Brilliant Staff

If the sequence $\{a_n\}$ converges, what is the most we can say about how many subsequences of $\{a_n\}$ converge?

Finitely many of them converge Infinitely many of them converge All but finitely many of them converge All of them converge

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by Brilliant Staff

If the convergent sequence $a_n$ of positive numbers satisfies $\lim_{n\to \infty} a_{n^2}+a_n^2= 56,$ what is the value of $\lim_{n\to\infty} a_n$ ?

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by Brilliant Staff

Consider a sequence $a_n$ . If $a_n$ converges to $x,$ is it true that every subsequence of the $a_n$ has a further subsequence that converges to $x$ ?

No Yes

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by Brilliant Staff

If a sequence $a_n$ satisfies $\lim_{n\to\infty} a_{2n}=3$ , is it true that $\lim_{n\to\infty} a_{2n+1}=3$ as well?

No Yes

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by Brilliant Staff

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