11-12. Taylor and Maclaurin Polynomials
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-19.jpeg)
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-20.jpeg)
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-21.jpeg)
13. Sequences
Arithmetic Sequence:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-22.jpeg)
Geometric Sequence:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-23.jpeg)
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/Geometric-sequence-formula-and-ratio-300x163.jpg)
Convergence of Sequence:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-24.jpeg)
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/Sequential-Convergence-e1619437755783.jpg)
14. Infinite Series
Infinite Sequence:
If a sequence is a list of numbers: … then a series is just the sum of the terms in the series: …
Infinite Series:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-25.jpeg)
Geometric Series:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-27.jpeg)
Partial Sum of Geometric Series:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/Geometric-series-partial-sum.jpg)
Converging and Diverging Series:
For the infinite series ![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/Sum-of-the-series.jpg)
the nth partial sum is given by ![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/nth-partial-sum.jpg)
If the sequence of partial sums converges to L,
then the series converges where L is the sum of the series.
If
diverges, then the series diverges.
Divergent Test for a Series:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/Series.jpg)
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/Divergent-Test-sentences-1.jpg)
Telescoping Infinite Series:
The telescoping series is of the form
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/Telescoping-series.jpg)
The series will only converge if and only if
approaches a finite number as n approaches infinity.
15. The Divergence and Integral Tests:
Integral Test:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-29.jpeg)
- The interval does not always need to start at 1.
- The function does not necessarily always need to be decreasing. It needs to decrease for the x-value larger than 1.
P-Series Test:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-31.jpeg)
Divergence Test:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-33.jpeg)
16. Comparison Tests:
Comparison Test:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/Comparison-Test.jpg)
Limit Comparison Test:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-35.jpeg)
17. Alternating Series Test:
Alternating Series Test:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-37.jpeg)
The alternating series test applies to the alternating harmonic series
because the individual terms satisfy the three conditions:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-38.jpeg)
So the alternating harmonic series
converges by the alternating series test.
Absolute and Conditional Convergence:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-39.jpeg)
This means that there are three possibilities for any given series: the series either converges absolutely or conditionally, or the series diverges.
18. Ratio and Root Test:
Ratio Test:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-40.jpeg)
Root Test:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-43.jpeg)
19. Power Series and Functions & Properties of Power Series:
Power Series:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-20.png)
Interval of Convergence:
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-46.jpeg)
20. Taylor and Maclaurin Series & Working with Taylor Series:
Taylor Series
![](https://openlab.citytech.cuny.edu/https-openlabcitytechcunyedu-poiriermat1575spring2021/files/2021/04/word-image-47.jpeg)
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