yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Worked example: alternating series | Series | AP Calculus BC | Khan Academy


2m read
·Nov 11, 2024

What are all positive values of P such that the series converges?

So let's see, we have the sum from n equal 1 to infinity of ((-1)^{n + 1} \frac{p}{6^{n}}).

There's a couple of things that might jump out at you. This ((-1)^{n + 1}) as (n) goes from 1 to 2 to 3, this is just going to alternate between positive 1, negative 1, positive 1, negative 1. So we're going to have alternating signs, so that might be a little bit of a clue of what's going on.

Actually, let's just write it out. This is going to be

  • when (n = 1), this is going to be (1^{2}), so it's going to be positive 1, so it's going to be (\frac{p}{6});
  • when (n = 2), this is going to be (1^{3}), so it's going to be minus (\frac{p}{6^{2}});
  • then plus (\frac{p}{6^{3}});
  • and I could even write (\frac{p}{6^{1}}) right over here;
  • then minus (\frac{p}{6^{4}})
  • and we're going to just keep going plus minus on and on and on and on forever.

So this is clearly a classic alternating series right over here. We can actually apply our alternating series test. Our alternating series test tells us that if this part of our expression, the part that is not alternating in sign, I guess you could say, if this part of the expression is monotonically decreasing, which is just a fancy way of saying that each successive term is less than the term before it.

And if we also know that the limit of this as (n) approaches infinity, that also has to be equal to zero. So the limit as (n) approaches infinity of (\frac{p}{6^{n}}) also has to be equal to zero.

So under what conditions is that going to be true? Well, to meet either one of those conditions, (\frac{p}{6}) has to be less than 1. If (\frac{p}{6}) was equal to 1, if for example (P) was 6, well then we wouldn't be monotonically decreasing. Every term here would just be one. It would be (1^{1}), (1^{2}), and on and on and on.

And if (p) is greater than 6, well then every time we multiply by (\frac{p}{6}) again we would get a larger number over and over again, and the limit for sure would not be equal to zero.

So we could say (\frac{p}{6}) needs to be less than 1. Multiply both sides by 6 and you get (P) needs to be less than 6.

They told us for what are all the positive values of (P). So we also know that (P) has to be greater than zero. Therefore, (p) is greater than zero and less than six, which is that choice right over here.

Once again, we're not going to say less than or equal to six, because if (P) was equal to six, this term is going to be (1^{n}) and so we're just going to have this. Would be one, this would be one. It would be 1 minus 1 plus 1 and on and on and on forever.

So definitely like that first choice.

More Articles

View All
Which mechanical keyboard should you buy -The most aesthetic keyboard ever
So let’s start with the conclusion. Should you buy the IQNEX F96 mechanical keyboard? Even though they start from 199 dollars to 250 dollars, the answer is yes! I spent so freaking much time using keyboards, sitting on my desk, writing with my keyboard. S…
Mr. Freeman, part 57
I invite you to play the game. Let us not give a damn about your IQ for a minute and go to the depths of imagination. Look closer. Assume that there’s some kind of time shift, and you’re suddenly went thousands of years back in time. What you got with you…
Rewriting expressions with exponents challenge 1 | Algebra 1 (TX TEKS) | Khan Academy
So we have this pretty complicated, some would say hairy, expression right over here. What I want you to do is pause this video and see if you can simplify this based on what you know about exponent rules. All right, now let’s do this together. There’s m…
Interviewing a Former White Nationalist | Trafficked with Mariana van Zeller
You’ll never get the truth from a current extremist. Their whole job is to lie to you and to spin things their own way. Which is why I say if you want the truth, talk to a former extremist. You still have the jacket? Still have the jacket? Oh, so this wa…
What’s the most effective way to offset the depreciation of your jet?
So what’s kind of the sweet spot in terms of how old the jet is where someone else is taking the depreciation and the big hit for you, but you’re not going to be stuck with something no one wants in 5 to 10 years? No, it’s a great question, Preo, because…
Worked Phillips curves free response question
Assume that the United States economy is currently in a short run equilibrium with the actual unemployment rate above the natural rate of unemployment. Part A says draw a single correctly labeled graph with both the long run Phillips curve and the short …