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Calculus

Introduction to sequences- Home
- Integral Calculus
- Sequence and Series

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Calculus

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Calculus

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In this section, we will be talking about monotonic and bounded sequences. We will learn that monotonic sequences are sequences which constantly increase or constantly decrease. We also learn that a sequence is bounded above if the sequence has a maximum value, and is bounded below if the sequence has a minimum value. Of course, sequences can be both bounded above and below. Lastly, we will take a look at applying theorem 7, which will help us determine if the sequence is convergent. One important to note from the theorem is that even if theorem 7 does not apply to the sequence, there is a possibility that the sequence is convergent. It's just that the theorem will not be able to show it.

Basic Concepts: Introduction to sequences

Note

**Theorems: **

1. A sequence is**increasing** if $a_n$ < $a_{n+1}$ for every $n \geq 1$.

2. A sequence is**decreasing** if $a_n$ > $a_{n+1}$ for every $n \geq 1$.

3. If a sequence is**increasing** or **decreasing**, then we call it **monotonic**.

4. A sequence is**bounded above** if there exists a number N such that $a_n \leq N$ for every $n \geq 1$.

5. A sequence is**bounded below** if there exists a number M such that $a_n \geq M$ for every $n \geq 1$.

6. A sequence is**bounded** if it is both **bounded above** and **bounded below**.

7. If the sequence is both**monotonic** and **bounded**, then it is always convergent.

1. A sequence is

2. A sequence is

3. If a sequence is

4. A sequence is

5. A sequence is

6. A sequence is

7. If the sequence is both

- IntroductionOverview:

a)Monotonic Sequencesb)Bounded Sequences - 1.
**Difference between monotonic and non-monotonic sequences**

Show that the following sequences is monotonic. Is it an increasing or decreasing sequence?a){$n^2$}b)$a_n= \frac{1}{3^n}$c)$\{\frac{n}{n+1}\}_{n=1}^{\infty}$d){1, 1.5, 2, 2.5, 3, 3.5, ...} - 2.
**Difference between bounded, bounded above, and bounded below**

Determine whether the sequences are bounded below, bounded above, both, or neithera)$a_n=n(-1)^n$b)$a_n=\frac{(-1)^n}{n^2}$c)$a_n=n^3$d)$a_n=-n^4$ - 3.
**Convegence of sequences**

Are the following sequences convergent according to theorem 7?a)$\{\frac{3}{n^3}\}_{n=1}^{\infty}$b)$\{\frac{(-1)^{2n+1}}{2}\}_{n=1}^{\infty}$c)$\{\sqrt{n}\}_{n=4}^{\infty}$

5.

Sequence and Series

5.1

Introduction to sequences

5.2

Monotonic and bounded sequences

5.3

Introduction to infinite series

5.4

Convergence and divergence of normal infinite series

5.5

Convergence & divergence of geometric series

5.6

Convergence & divergence of telescoping series

5.7

Divergence of harmonic series

5.8

P Series

5.9

Alternating series test

5.10

Divergence test

5.11

Comparison & limit comparison test

5.12

Integral test

5.13

Ratio test

5.14

Root test

5.15

Absolute & conditional convergence

5.16

Radius and interval of convergence with power series

5.17

Functions expressed as power series

5.18

Taylor series and Maclaurin series

5.19

Approximating functions with Taylor polynomials and error bounds

We have over 170 practice questions in Integral Calculus for you to master.

Get Started Now5.1

Introduction to sequences

5.2

Monotonic and bounded sequences

5.4

Convergence and divergence of normal infinite series

5.5

Convergence & divergence of geometric series

5.6

Convergence & divergence of telescoping series

5.7

Divergence of harmonic series

5.8

P Series

5.9

Alternating series test

5.10

Divergence test

5.11

Comparison & limit comparison test

5.12

Integral test

5.13

Ratio test

5.14

Root test

5.15

Absolute & conditional convergence

5.16

Radius and interval of convergence with power series

5.17

Functions expressed as power series

5.18

Taylor series and Maclaurin series