Converging And Diverging Series

Correct Answer: Converge.
1. Since the nth term divergence test does not work, the easiest way to do this is by the Direct Comparison Test:
8/(n^2)> 8/(3n^2+2n+1)
2. P-Series:
8/n^2 and 2>1 so 8/n^2 converges
3. Since 8/n^2 converges so does the given series since it is smaller

Wrong Answers:
Diverge. There are not too many ways to come up with this conclusion. However a common mistake is that when you use any type of comparison test, you must make sure you are comparing a series to the same degree. For example in this problem if you compared the series to 8/n rather than 8/(n^2) you would have come that 8/n diverges. And even though that is true you may also make the mistake of saying the the given series also diverges, although you cannot compare those two series because the greater diverges, so the smaller could converge or diverge. However both of those mistake are fairly common.

Cannot Be Determined. You may conclude that it cannot be determine because there are restrictions on the direct comparison test (as I just described above). For example, if some series "a" diverges, and a>b then you could not conclude that series "b" also diverges. However, that rule does not apply for converging series, but the rule is in fact the opposite. It is a common mistake to mix up those rules for comparing converging and diverging series.

Correct Answer: Diverge.
1. Do the nth Term Divergence Test and use L'Hopitals Rule
lim 3n^3/(5n^3+9) = lim 9n^2/(15n^2) = lim 18n/(30n)= 18/30
n->∞ n->∞ n->∞

3/5 does not equal zero, so the series diverges.

Wrong Answers:
A. If someone does the nth term divergence test then he/she would get infinite/infinite. Rather than using L'Hopitals rule, one may assume that the series is inconclusive.

B. There are a couple of ways one may conclude this diverges. First someone may look at the first few terms of the series and see that it id decreasing. And while the sequence does converge, the sequence of partial sums does not. Another way to get this wrong answer is if someone does the nth term divergence test and gets 3/5, then he/she may think it is what the series converges to, or that since 3/5

Correct Answer: 2 Only.
According to the P-Series Test, if you have
∞
∑ #/(n^p) and p is less than or equal to 1, the series diverges, and if p is greater than 1 it conv.
n=1
50>1 thus the series converges
Wrong Answers:
A. if this series is evaluated using a p-series test, then remember if p is less than or equal to one then the series divergres. People often forget what happens when p=1.

B. It cannot be option 2 either according to the p-series test as I just described above, since 1/50

Correct Answer: Conditionally Converges.
First take the absolute value of the series. Then find whether the series converges or diverges, to do this I used the Integral test since a(n) decreases as n increases.When I did this, I got it diverges. So you must next use the Conditional Convergence Test. When I did this I found that a(n) decreases to zero as n increases, so it conditionally converges.

Wrong Answers:
While Converging would be a less common incorrect answer, Diverging may be be pretty common because people may forget to check for conditional convergence.

Correct Answer:Diverges.
1. Use the Root Test.
lim (-7/n) +8 =0+8=8
n->∞
8>1 so the series diverges.

Wrong Answers:
B. If someone thought the lim as n-> infinite of -7/n=-7 then they would get 1, and according to the root test if L=1 then the test is inconclusive.
C. some one may have mixed this up with the p-series test where if n>1, then the series converges. Another mistake that may be made is that the sequence of numbers gets smaller as n gets bigger, so that may lead to the faulty conclusion that the series converges, however the partial sum of the series always increases because of the 8's presence.

Correct Answer:
-6 is less than x which is less than -4
1. Use the Ratio Test to Find the Interval
∑ ((5n!)(x+5)^n+7)/n= ((5(n+1)!(x+5)^n+8)/n+1 x n/((5n!)(x+5)^n+7))= x+5; -1