True/False: Determine whether each of the statements that...

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: If the tail of a sequence converges, the sequence converges. (b) True or False: If $\left\{a_{k}\right\}$ and $\left\{b_{k}\right\}$ are two divergent sequences, then the sequence $\left\{a_{k}+b_{k}\right\}$ diverges. (c) True or False: If $\left\{a_{k}\right\}$ is a convergent sequence of rational numbers, it must converge to a rational number. (d) True or False: Every convergent sequence is bounded. (e) True or False: Every bounded sequence is convergent. (f) True or False: $\lim _{k \rightarrow \infty} \frac{k^{1000000}}{(1.000001)^{k}}=0$ (g) True or False: Every increasing sequence of negative numbers converges. (h) True or False: Every increasing sequence of positive numbers converges.

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: If the tail of a sequence converges, the sequence converges.
(b) True or False: If $\left\{a_{k}\right\}$ and $\left\{b_{k}\right\}$ are two divergent sequences, then the sequence $\left\{a_{k}+b_{k}\right\}$ diverges.
(c) True or False: If $\left\{a_{k}\right\}$ is a convergent sequence of rational numbers, it must converge to a rational number.
(d) True or False: Every convergent sequence is bounded.
(e) True or False: Every bounded sequence is convergent.
(f) True or False: $\lim _{k \rightarrow \infty} \frac{k^{1000000}}{(1.000001)^{k}}=0$
(g) True or False: Every increasing sequence of negative numbers converges.
(h) True or False: Every increasing sequence of positive numbers converges.

Key Concepts

Rate of Growth: Exponential vs Polynomial

This concept explains that exponential functions grow (or decay) much faster than polynomial functions. In limits, if the denominator of a sequence is exponential and the numerator is a polynomial, the exponential term will dominate as the index tends to infinity, often forcing the limit to zero.

Monotonic Sequence Convergence

Monotonic sequences, those that are either entirely non-increasing or non-decreasing, have special convergence properties. If a monotonic sequence is also bounded, the Monotone Convergence Theorem ensures that it converges. This concept is key in understanding why certain sequences, depending on their monotonicity and bounds, converge automatically.

Bounded Sequences

A bounded sequence is one in which all terms lie within some fixed interval. Convergence always implies boundedness since a convergent sequence cannot stray arbitrarily far from its limit. However, being bounded does not guarantee convergence, as there may be oscillation or chaotic behavior within the limits.

Tail of a Sequence

The tail of a sequence means all the terms of the sequence from a certain point onward. In convergence, only the behavior of the tail is significant because finite changes in the beginning of the sequence do not affect whether or not the sequence converges.

Divergent Sequences and Cancellation

This concept addresses the behavior of sequences that do not settle to a limit and considers whether combining them (for instance, summing two divergent sequences) might result in cancellation effects that lead to convergence. It underlines the fact that divergence is not necessarily preserved under term-by-term addition.

Sequence Convergence

This concept refers to the idea that a sequence approaches a specific number, called its limit, as its index increases indefinitely. The formal definition involves the property that for every positive number ?, there exists an index N such that the distance between any term after N and the limit is less than ?.

Rational and Irrational Limits

A sequence consisting entirely of rational numbers may still converge to an irrational number. This is because the rational numbers are dense in the real numbers, allowing a sequence of rationals to approach any real limit despite every individual term being rational.

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