I’m getting more and more frustrated as the weeks go on as I still can’t visualize what’s happening with series. Whenever I feel like I’m making progress and starting to get a handle on what’s happening, I watch another video and everything I think I understand falls apart. It feels like I’m trying to juggle 25 things in my brain at once but I can only manage a few of them at a time. I started this unit in Week 166 so it’s been ~2.5 months of working through these concepts and yet I feel like I’ve barely made any progress. I did make it through the single exercise from the section Maclaurin Series of e^x, sin(x) and cos(x) this week and watched the first 4 videos in the following section Representing Functions as Power Series, but I didn’t really understand what I watched… So (once again) I’m not going to try to explain much of what I learned this week in this blog post because I can’t. 😡

To get through the exercise from Maclaurin Series of e^x, sin(x) and cos(x), I had to more-or-less memorize the series’ formulas and patterns for e^x, sin(x) and cos(x). They are:

• e^x = _n=0Σ^∞ xⁿ/n!
  • = {1 + x + (x²/2!) + (x³/3!) + (x⁴/4!) + … }
    
• sin(x) = _n=0Σ^∞ (-1)ⁿ⁺¹ * x²ⁿ⁺¹/(2n + 1)!
  • = {x – (x³/3!) + (x⁵/5!) – (x⁷/7!) + (x⁹/9!) – …}
    
• cos(x) = _n=0Σ^∞ (-1)ⁿ⁺¹ * x²ⁿ/(2n)!
  • = {1 – (x²/2!) + (x⁴/4!) – (x⁶/6!) + (x⁸/8!) – …}

(As a side note to highlight how much I don’t understand what’s going on, I’ve been working through these types of series for a few weeks [months?] and just had to look up the a_n formulas for both sin(x) and cos(x)and don’t understand why they look the way they do or how/why they work… So that’s cool.)

Having the patterns of each of those series somewhat memorized, the exercise would give me a series that looked similar to one of those three and I had to use some sort of algebra to the series to come up with whatever solution the question was looking for. Here are 3 questions I worked though:

Question 1

In this question, the formula a_n is x²ⁿ/n! which looks very similar to the series formula for e^x which is xⁿ/n!. Recognizing that and understanding that the series _n=0Σ^∞ xⁿ/n! literally IS e^x, you then input x² into x in the function e^x and then, since the question is asking to find the solution to f((ln(3))^1/2), you input x = (ln(3))^1/2 into e^{x^2} and solve.

Question

Question 3

This question is a good example of what I meant when I said I had to have the patterns for each of the series memorized. The given series-pattern is exactly the same as it is for sin(x) except that x is replaced with π. To solve, you just input π into the sin function.

I got through that exercise on Tuesday and found it pretty easy, although I didn’t fully understand what was happening. I then started the following section, Representing Functions as Power Series, on Wednesday. Here are the notes I took from the first video:

Looking back on this note, I don’t know now what I was talking about then. I think what the video was explaining is that if a function can be represented by a series (which doesn’t make sense to me), you can figure out what the definite integral of the function would be by integrating the series’ formula, evaluating the antiderivative of the formula with the given bounds, and then, if the result is a formula for a geometric series (which in this example it is as it turns into (1/4)ⁿ⁺¹), you can then input the new formula into the a/(1 – r) formula to find out what the series converges on which (I think) is the ‘area’ underneath the curve of the function from the very start.

(I have absolutely NO clue what I’m talking about… 😭)

The second video was along the same lines to the first. Here are my notes from that vid:

The first part of this video (which is shown in the first screen shot above) explained that if you want to find the third derivative of a series, you can write out the terms of the series and then find the 1^st, 2^nd and 3^rd derivatives of each term. The question asked to find what the third derivative of the function evaluated at x = 0 was which, as you can see, was equal to 6. This was because as you used the power rule, the degree of the first term eventually got knocked down to 0, i.e. it became 6x⁰, and so evaluating 6x⁰ at x = 0 equals 6. The second half of the video goes on to show that instead of finding the 1^st, 2^nd and 3^rd derivatives of each term individually, you can find the 1^st, 2^nd and 3^rd derivatives of the formula, a_n, and then evaluate the formula at x = 0. One thing to keep in mind is that when evaluating that 3^rd derivative of a_n, the only term that needs to be inputted (i.e. the only n that the 3^rd derivative a_n needs to be evaluated at) is 0 since the last factor in a_n is x²ⁿ = (0)²ⁿ and so any n other than 0 will result in (0)²ⁿ = 0 as opposed to (0)²⁽⁰⁾ = 1… 

(…This last sentence if very confusing. 😵‍💫)

As bummed out as I am that I don’t know what’s going on with series, I’m still confident I’ll eventually get there. I have a feeling that even when I get through Series (1,440/2,000 M.P.), it’s going to be a few months (or years?) before I’m able to visualize >90% of what’s going on. I’m definitely demoralized AF right now but I’ve been here before and I know I’ll eventually through it. You might not believe me, but I’m series-ous.