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Math 126
Sequences and Series Review Sheet
This sheet is intended to help you study for the final exam. In particular, it will list out strategies for sequences and series, and help you prioritize what is most important to memorize, so you don’t have to worry about cramming the whole book into your head. Unfortunately, we have agreed that there will be NO CHEATSHEETS FOR THE FINAL, so you will have to memorize things.
1 Sequences
A sequence is just a list of numbers
a 1 , a 2 , a 3 ,...
There are only a couple of ways to check if a sequence converges. The three techniques you should know are:
- Relate it to a function so that an = f (n) and take the limit limx→∞ f (x).
- Use the monotonic convergence theorem.
- Use the squeeze theorem.
2 Series
A series is a sum of numbers
S =
∑^ ∞
i=
an.
The nth partial sum is
sn =
∑^ n
i=
an.
A series converges if and only if its SEQUENCE of partial sums converges. Now, here is a strategy list for checking to see if a Series
i=1 an^ converges.
- First try the Divergence Test, or “nth term test” to make sure an → 0. If it fails this test, you are done.
- Is there a (−1)n? Then its probably an alternating series, so use the alternating series test. Its easy, except sometimes annoying to show that the terms are decreasing.
- p-series:
n=
1 np^ converges for^ p >^ 1 only.
- Geometric series
arn−^1 converge for |r| < 1 only, in which case, the sum is (^1) −ar. Remember it is easy to find a and r: a is the first term in the series, and r is the ratio.
- Telescoping series look like partial fractions problems, for example:
n^2 − 1.
- If none of these things work, try comparing to a p-series or geometric series. Remem- ber, you need to have positive terms for the comparison tests.
- And of course, don’t forget the Ratio test and Root test, which determine when a series is absolutely convergent (meaning, the sum
|an| converges). They work almost exactly the same, both requiring a value < 1 for absolute convergence, but they are useful on different series. The Ratio test is good when you have n! or a constant to a power like 7n, but the Root test is good when you have a complicated expression to a power like (1 + n)n.
- The Integral Test should be used on anything that looks more fiendly when you replace the ‘
’ with a ‘
’. Classic examples involve the natural log, for example:
∑ (^) (ln(n))k n. A useful thing to remember is that YOU CAN ONLY FIND THE VALUE OF A SERIES USING 4, OR 5 ABOVE.
3 Power Series
You MUST know the formula for the Taylor series for a function f (x) centered at x = a:
∑ cn(x − a)n, cn =
f (n)(a) n!
You should memorize, or at least be very familiar with, the Maclaurin series for the following functions:
ex, sin(x), cos(x),
1 − x
, ln(1 − x), (1 + x)k, tan−^1 (x)
Interval of Convergence. For power series, ONE OF THE MOST IMPORTANT PROB- LEMS is to find the interval of convergence. For a series centered at x = a, it will look something like [a − R, a + R], [a − R, a + R), etc., where R is the radius of convergence. For this use either the Root test or the Ratio test to find R, then test the endpoints (memorize this as “R or R for R”). Often, one of the endpoints will be an alternating series. Finally, in using power series for approximating the value of some function or integral, say (^) ∫ 1
0
e−x
2 , or
you have two options:
- If you get an alternating series (like the first example), life is good: find the term which is smaller than your desired error and cut off right before that term.
- If you don’t get an alternating series (like the second example), life is not so good, you have to use Taylor’s formula. Remember, it looks like the next term in the Taylor series: |Rn(x)| ≤
|f (n+1)(z)| (n + 1)!
|x − a|n+
where z is some point in the interval that you care about centered at a. Like the error problems for numerical integration (which is not on the final), THE TRICKY PART IS FIGURING OUT WHICH z TO USE! Usually, it will be an endpoint point which gives you the biggest value of f (n+1). For example, in approximating
99 .9, use a Taylor series centered at x = 100 and look at the smallest interval centered at 100 containing 99.9 and choose z which gives the biggest value of f (n+1)^ on this interval. In this case, you use the endpoint z = 99. 9 ∈ (99. 9 , 100 .1).
