5 Sequences Free Online FlashCards

State the Monotone Convergence Theorem.

Every increasing sequence bounded above converges, and every decreasing sequence bounded below converges.

Why is a fixed-point equation not enough?

Solving the fixed-point equation produces only possible limits. Convergence must be proved separately, typically using monotonicity and an appropriate bound.

What effect do finitely many initial terms have?

Changing or adding finitely many initial terms does not affect whether a sequence converges or the value of its finite limit.

What is a sequence?

A sequence is an ordered list of real numbers indexed by the positive integers, written $\{a_n\}_{n=1}^{\infty}$.

What does $a_n$ represent?

The term $a_n$ is the term at index $n$, and $n$ must be an integer even when the related function accepts real inputs.

What is the formal definition of $a_n\to L$?

For every $\varepsilon>0$, there is an integer $N$ such that $|a_n-L|<\varepsilon$ whenever $n\ge N$.

Why does $(-1)^n$ diverge?

A sequence diverges when no finite real number satisfies the definition of convergence. For example, $(-1)^n$ diverges because it alternates between $-1$ and $1$.

How can a function limit determine a sequence limit?

If $a_n=f(n)$ and $\lim_{x\to\infty}f(x)=L$, then $\lim_{n\to\infty}a_n=L$.

When does the quotient limit law apply?

For $a_n\to L$ and $b_n\to M$, the quotient limit is $\frac{L}{M}$, provided $M\ne0$.

State the Squeeze Theorem for sequences.

If $b_n\le a_n\le c_n$ eventually and both outside sequences approach $L$, then $a_n\to L$.

How can differences test sequence monotonicity?

To test monotonicity, examine $a_{n+1}-a_n$. If it is always nonnegative, the sequence is nondecreasing; if always nonpositive, it is nonincreasing.

What does it mean for a sequence to be bounded?

A sequence is bounded above if $a_n\le U$ for every $n$, bounded below if $a_n\ge L$, and bounded if both conditions hold.