Notes 03-30-09

Today we continued with Power Series and focused on two particular types: Taylor Series and Maclaurin Series. We first defined the two sub-series, then particed constructing them given different conditions.

Power Series

Today in class, we learned about power series and how to write them. Some power series are centered at x=0. These series are defined as the infinite sum of A sub n (An) multiplied by x^n. Other series are centered at x=c and these are defined as the infinite sum of the An multiplied by (x-c)^n. When writing a power series we first make sure the given function is in the geometric series form of 1st term/ (1-ratio). Then, the rest is easy...you just write the first term in the series and multiply each of the following terms by the ratio. Remember, the absolute value of the ratio must be less than one and to find the values of the interval of convergence for the power series just solve that absolute value equation.

Pythagorean Tree

This is the video we recorded in class today showing the Pythagorean Tree fractal.

(Note that clicking on the link will take you away from the blog...)

Notes from Class March 23-7th Period

Today we learned the Ratio Test. We were able to determine convergence or divergence of a series by dividing its sequence of a(n+1) over a(n). We also derived e by using the Ratio Test. Notice how the Ratio Test is similar to the Geometric Series Rule (r in geometric series is term/previous term).

Idea of Partial Sums

I'm posting a video of the partial sums graphs that we looked at in class before spring break. This particular post may or may not prove useful, but I am trying out the idea of making and posting a video.

Partial sums video

(still working on embedding...)

Comparison Tests

Today in class (Wednesday 3/11), we learned about how to determine if an alternating series converges (conditionally or absolutely) or not. An alternating series is most easily recognized by (-1)^n involved somewhere, which will alternately change the values of the sequence from positive to negative. We learned some cool stuff about what it means for a series to converge as well, and discovered that certain sums can be different values depending on the order of the terms.

On March 9th in class, we discussed a new way to see if summations are converging or diverging. We now know that if the definite integral from 1 to infinity converges, then so does the respective summation. We also learned the P-Series Test, which is very specific in its application; it can only be used for functions that are 1/n^p.

In class March 5th we discussed sequences. We talked about both geometric and arithmetic equations and how to solve them. We also discussed combinations and permutations. And, we mentioned convergence and divergence and how to determine whether something converges or diverges.

Arc Length of a Polar Curve

Today we learned how to derive and find the arc length of a polar curve.
We also learned how the arc length formula of a parametric function can be applied to find the arc length of a rectangular curve.