Chapter 7: Sequences and Series

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Every sequence is a function.

(b) True or False: The third term of the sequence \(\left \{ k+1 \right \}_{k=1}^{\infty }\) is \(4\).

(c) True or False: The third term of the sequence \(\left \{ k^2 \right \}_{k=2}^{\infty }\) is \(9\).

(d) True or False: Every sequence of real numbers is either increasing or decreasing.

(e) True or False: Every sequence of numbers has a smallest term.

(f) True or False: Every recursively defined sequence has an infinite number of distinct outputs.

(g) True or False: Every sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

(h) True or False: Every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

A strictly increasing sequence of partial sums: if \[ a_k > 0\] for \[k \;\epsilon\; \mathbb{Z}^{+}\]

Explain why the sequence of partial sums for the series \[\sum_{k=1}^{\infty}a_{k}\] is strictly increasing.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A series that converges absolutely.

(b) A series that converges conditionally.

(c) A series \[\sum _{k=1}^{\infty}a_{k}\] such that \[\lim_{k \to \infty} \left | \frac{a_{k+1}}{a_{k}} \right | =1\] but the series is absolutely convergent.

A bounded sequence of partial sums: If the series \[\sum_{k=1}^{\infty}a_{k}\] converges, explain why the sequence of partial sums is bounded.

Find the values of p that make the series converge absolutely, the values that make the series converge conditionally, and the values that make the series diverge. \[\sum_{k=0}^{\infty}(-1)^{k}k^{p}e^{-k^{2}}\]

ddd

1. True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(A) True or False:$\sum _{k=1}^{\infty }{(-1)}^{k+1}{a}_k$is an alternating series.

Functions: Provide definitions for each of the following:

• function
• the domain of a function
• the codomain of a function
• the function $f$is increasing on interval localid="1654357677089" $\left[a,b\right]$
• the function localid="1654357683495" $f$is strictly increasing on interval localid="1654357689590" $\left[a,b\right]$
• the function localid="1654357698687" $f$is decreasing on interval localid="1654357702734" $\left[a,b\right]$
• the function localid="1654357731613" $f$is strictly decreasing on interval localid="1654357708010" $\left[a,b\right]$
• the function localid="1654357726536" $f$is constant on interval localid="1654357712112" $\left[a,b\right]$
• the function localid="1654357721905" $f$is bounded on intervallocalid="1654357717212" $\left[a,b\right]$

Read the section and make your own summary of the material.

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