# Thread: Sequences and their Limits

1. ## Sequences and their Limits

Use Theorem 3 to show that the sequence 0.7, 0.77, 0.777, 0.7777, 0.77777, 0.777777, 0.7777777, ... has a limit.

Theorem 3 is ...

Suppose that the sequence $\displaystyle a_{k}$ is nondecresing and bounded above by a number A. That is,

$\displaystyle a_{1} \leq a_{2} \leq a_{3} \leq A$

Then $\displaystyle a_{k}$ converges to some finite limit a, with a $\displaystyle \leq A$. Similarly, if $\displaystyle b_{k}$ is nonincreasing and bounded below by a number B, then $\displaystyle b_{k}$ converges to a finite limit $\displaystyle b \geq B$.

edit: How do I do the less than or equal to sign in LaTeX?

2. Originally Posted by larson
Use Theorem 3 to show that the sequence 0.7, 0.77, 0.777, 0.7777, 0.77777, 0.777777, 0.7777777, ... has a limit.

Theorem 3 is ...

Suppose that the sequence $\displaystyle a_{k}$ is nondecresing and bounded above by a number A. That is,

$\displaystyle a_{1}$ <, $\displaystyle a_{2}$ < $\displaystyle a_{3}$ < A

Then $\displaystyle a_{k}$ converges to some finite limit a, with a < A. Similarly, if $\displaystyle b_{k}$ is nonincreasing and bounded below by a number B, then $\displaystyle b_{k}$ converges to a finite limit b > B.

edit: How do I do the less than or equal to sign in LaTeX?
$\displaystyle /leq$ gives $\displaystyle \leq$ just change / to \

3. Originally Posted by larson
Use Theorem 3 to show that the sequence 0.7, 0.77, 0.777, 0.7777, 0.77777, 0.777777, 0.7777777, ... has a limit.

Theorem 3 is ...

Suppose that the sequence $\displaystyle a_{k}$ is nondecresing and bounded above by a number A. That is,

$\displaystyle a_{1}$ <, $\displaystyle a_{2}$ < $\displaystyle a_{3}$ < A

Then $\displaystyle a_{k}$ converges to some finite limit a, with a < A. Similarly, if $\displaystyle b_{k}$ is nonincreasing and bounded below by a number B, then $\displaystyle b_{k}$ converges to a finite limit b > B.

edit: How do I do the less than or equal to sign in LaTeX?
you can use the first part of the theorem with $\displaystyle A = 0. \bar{7}$ or $\displaystyle A = 0.8$

4. Originally Posted by larson
Use Theorem 3 to show that the sequence 0.7, 0.77, 0.777, 0.7777, 0.77777, 0.777777, 0.7777777, ... has a limit.

Theorem 3 is ...

Suppose that the sequence $\displaystyle a_{k}$ is nondecresing and bounded above by a number A. That is,

$\displaystyle a_{1}$ <, $\displaystyle a_{2}$ < $\displaystyle a_{3}$ < A

Then $\displaystyle a_{k}$ converges to some finite limit a, with a < A. Similarly, if $\displaystyle b_{k}$ is nonincreasing and bounded below by a number B, then $\displaystyle b_{k}$ converges to a finite limit b > B.

edit: How do I do the less than or equal to sign in LaTeX?
and I will give you a hint $\displaystyle a_n=7\sum_{t=1}^{n}\cdot\bigg(\frac{1}{10}\bigg)^{ t}$

5. Originally Posted by Mathstud28
$\displaystyle /leq$ gives $\displaystyle \leq$ just change / to \
it suffices to type le, likewise, ge and ne for greater thank and not equal to respectively

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# solve geometric progressions of 0.7 0.77 0.777

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