Category: Week 12 group post (Page 1 of 4)

11-12. Taylor and Maclaurin Polynomials

13. Sequences

Arithmetic Sequence:

Geometric Sequence:

Convergence of Sequence:

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:

Geometric Series:

Partial Sum of Geometric Series:

Converging and Diverging Series:

For the infinite series

the nth partial sum is given by

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:

Telescoping Infinite Series:

The telescoping series is of the form

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:

• 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:

Divergence Test:

16. Comparison Tests:

Comparison Test:

Limit Comparison Test:

17. Alternating Series Test:

Alternating Series Test:

The alternating series test applies to the alternating harmonic series because the individual terms satisfy the three conditions:

So the alternating harmonic series   converges by the alternating series test.

Absolute and Conditional Convergence:

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:

Root Test:

19. Power Series and Functions & Properties of Power Series:

Power Series:

Interval of Convergence:

20. Taylor and Maclaurin Series & Working with Taylor Series:

Taylor Series