Quick Calc. II tips needed

[Saifa]

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Final is tomorrow morning, need at least a 60 to come out of this ****show with a B. A significant portion of the exam will be assessing convergence of series. The final weeks of this class were especially nebulous and hard to reach for me. I have your typical tool-box of convergence tests but need some advice on knowing which ones to pull out. Some are simple, like p-series and geometric, but many others are way more complex and intimidating, especially when being faced with a random summation. Can anyone offer any hints? I've been on Paul's notes and Khan academy all night but could always go for some extra wisdom.

 

[Lawpy]

Final is tomorrow morning, need at least a 60 to come out of this ****show with a B. A significant portion of the exam will be assessing convergence of series. The final weeks of this class were especially nebulous and hard to reach for me. I have your typical tool-box of convergence tests but need some advice on knowing which ones to pull out. Some are simple, like p-series and geometric, but many others are way more complex and intimidating, especially when being faced with a random summation. Can anyone offer any hints? I've been on Paul's notes and Khan academy all night but could always go for some extra wisdom.

Paul's online math notes, Khan Academy, Visual Calculus and your calculus textbook are the go to resources to help you dominate calculus (both single and multivariate).

Regarding tests of convergence, I like to think of them in terms of a flowchart or some methodical approach like the following:

1. Treat the series as a sequence and see how the nth term fares at infinity. If the limit of the sequence ends up being anything but 0, the series diverges. So if i have a series like:

1/2 + 2/3 + 3/4 + 4/5 + ... n/(n+1), as n approaches infinity, the limit is 1, so the series diverges. Use this Divergence Test first to quickly determine if the series diverges.

2. If the limit of the sequence is 0, you need to use something else. Now, if you see a series that's alternating like:

1 - 1/2 + 1/3 - 1/4 + 1/5 ... you should use the Alternating Series Test. This includes seried having trig functions since they alternate signs. The test is simple: you treat the series as if all the terms are positive. If the limit of the sequence is 0 and the terms are decreasing, the series converges.

You can also use the Ratio and Root Tests for alternating series since they take the absolute value of the sequence terms.

3. To decide what other tests of convergence to use, I would use the Integral Test if you are dealing with a simple, decreasing sequence that can be easily integrated. Otherwise, you would have to compare a difficult series with a much simpler series that has a known convergence/divergence (Direct Comparison/Limit Comparison Tests)

4. And finally, if you are dealing with factorials, use the Ratio Test for quick calculation and make sure the result doesn't end up equal to 1 (test fails). And use the Root Test if you are dealing with complex exponential functions like n^n terms.

A lot of this is practice. You can read more here:

http://tutorial.math.lamar.edu/Classes/CalcII/SeriesStrategy.aspx

Hope this helps and good luck with the final

 

[wizzed101]

plug numbers in your calculator, 5 terms are usually more than enough then BS your way through the explanation. If the test is multiple choice then you simply cannot fail.

 

[Saifa]

Paul's online math notes, Khan Academy, Visual Calculus and your calculus textbook are the go to resources to help you dominate calculus (both single and multivariate).

Regarding tests of convergence, I like to think of them in terms of a flowchart or some methodical approach like the following:

1. Treat the series as a sequence and see how the nth term fares at infinity. If the limit of the sequence ends up being anything but 0, the series diverges. So if i have a series like:

1/2 + 2/3 + 3/4 + 4/5 + ... n/(n+1), as n approaches infinity, the limit is 1, so the series diverges. Use this Divergence Test first to quickly determine if the series diverges.

2. If the limit of the sequence is 0, you need to use something else. Now, if you see a series that's alternating like:

1 - 1/2 + 1/3 - 1/4 + 1/5 ... you should use the Alternating Series Test. This includes seried having trig functions since they alternate signs. The test is simple: you treat the series as if all the terms are positive. If the limit of the sequence is 0 and the terms are decreasing, the series converges.

You can also use the Ratio and Root Tests for alternating series since they take the absolute value of the sequence terms.

3. To decide what other tests of convergence to use, I would use the Integral Test if you are dealing with a simple, decreasing sequence that can be easily integrated. Otherwise, you would have to compare a difficult series with a much simpler series that has a known convergence/divergence (Direct Comparison/Limit Comparison Tests)

4. And finally, if you are dealing with factorials, use the Ratio Test for quick calculation and make sure the result doesn't end up equal to 1 (test fails). And use the Root Test if you are dealing with complex exponential functions like n^n terms.

A lot of this is practice. You can read more here:

http://tutorial.math.lamar.edu/Classes/CalcII/SeriesStrategy.aspx

Hope this helps and good luck with the final

Far better summary than my professor ever made, thank you!

If the test is multiple choice then you simply cannot fail.

The tests are hand-scrawled open-answer photocopied the day before, 10 points correct, 0 incorrect, 0 correct but work was bad (which will earn a red "?"), no questions asked. It's just been a peach of a time.

 

[Matthew9Thirtyfive]

Paul's Online Math Notes is your friend. Do some practice problems and focus on recognizing which test to use. Big hints:

1. A p-series will only converge if p > 1, and a geometric series will only converge if |r| < 1.

2. Use the comparison test if the series seems like a p-series or a geometric series; however don't **** up the rules! Just because the smaller series converges doesn't mean the larger series does! Also, the comparison test requires the series to be positive (eventually--if the first couple terms are negative, that's okay).

3. If there's a factorial, use the root test.

4. Don't be afraid of algebra. You may have to move some things around to get it in a recognizable state. This might make it easier to determine which test to use, e.g., writing it as a root.

5. If it's not easy to integrate, don't use the integral test. You'll waste a ton of time when you could use an easier test.

Just run through your check list:

1. Does it look like it does not converge to 0? Use the divergence test.

2. Is it p or geometric? Remember the rules. If it looks like one of those, try the comparison test.

3. If you can't tell, try some algebra to see if you can rewrite it in a recognizable state.
3a. Roots? Root test
3b. Factorial? Ratio test
3c. Alternating series? Alternating series test
3d. Easy integral? Integral test

4. ?????

5. Profit

 

[Saifa]

If it looks like one of those, try the comparison test.

This is the final test that I have really struggled to master, there seems to be a lot of arbitration when reasoning through it, but that is probably just my poor understanding.

 

[Matthew9Thirtyfive]

This is the final test that I have really struggled to master, there seems to be a lot of arbitration when reasoning through it, but that is probably just my poor understanding.

Personally, I think that if done properly, calc 2 should be the most difficult of the three. It forces you to recognize novel patterns and how to configure them into something you can work with. That is much more difficult that the straightforward problems in most calc 1 courses, and once you get through calc 2, calc 3 is easy. Just remember that everything can be boiled down to an algorithm. Just follow it, and you'll be good.

 

[Saifa]

Personally, I think that if done properly, calc 2 should be the most difficult of the three.

A well-echoed sentiment. As someone who pretty much dusted calc. I, I NEVER saw it coming and have been pretty humbled by the course. Many bright students at my school failed it their first time around and are coming back for their second final tomorrow. I feel more adaptable having gone through it, but I hope that merit holds up to GPA ground lost. You and other SDNer have to-date helped me with it tremendously.

 

[Matthew9Thirtyfive]

A well-echoed sentiment. As someone who pretty much dusted calc. I, I NEVER saw it coming and have been pretty humbled by the course. Many bright students at my school failed it their first time around and are coming back for their second final tomorrow. I feel more adaptable having gone through it, but I hope that merit holds up to GPA ground lost. You and other SDNer have to-date helped me with it tremendously.

Glad to be of service. I enjoyed my upper division courses more (especially real analysis 😍), but I'll always have a special place in my heart for calc.

 

[Saifa]

I think this was extremely helpful, might have even gotten an A! Was blind sided by a Taylor series problem that I didn't prepare for at all because we spent like ten minutes on it all year. Also didn't realize I had to use trig. sub. to solve an integral until only ten minutes remained. Solved it. It was pretty clutch.

 

[Lawpy]

I think this was extremely helpful, might have even gotten an A! Was blind sided by a Taylor series problem that I didn't prepare for at all because we spent like ten minutes on it all year. Also didn't realize I had to use trig. sub. to solve an integral until only ten minutes remained. Solved it. It was pretty clutch.

Congrats! Glad to hear

 

[Matthew9Thirtyfive]

I think this was extremely helpful, might have even gotten an A! Was blind sided by a Taylor series problem that I didn't prepare for at all because we spent like ten minutes on it all year. Also didn't realize I had to use trig. sub. to solve an integral until only ten minutes remained. Solved it. It was pretty clutch.

Glad to hear it. 🙂

 

[leonardoson]

Personally, I think that if done properly, calc 2 should be the most difficult of the three. It forces you to recognize novel patterns and how to configure them into something you can work with. That is much more difficult that the straightforward problems in most calc 1 courses, and once you get through calc 2, calc 3 is easy. Just remember that everything can be boiled down to an algorithm. Just follow it, and you'll be good.

Not if your calc 3 professor puts broken questions on the final.... a broken question that was on a review sheet. Then calc 3 is harder

 

[freak7]

Not if your calc 3 professor puts broken questions on the final.... a broken question that was on a review sheet. Then calc 3 is harder

"If done properly"