Submit neatly written solutions to the following questions. You do not have to t

Question

Submit neatly written solutions to the following questions. You do not have to type your solutions, just submit solutions that are clear, and easily read. Show sufficient work for full credit 1. Determine if the series S- X(-1)In converges absolutely, converges conditionally, or diverges. Indicate the test/s you used. 2. Determine if the series S - +4 COnVerges absolutely, converges conditionally, or di- n= 1 verges. Indicate the test/s you used cO 3. Determine if the series -? (-1)"n3??? converges absolutely, converges conditionally, or diverges. Indicate the test/s you used. 4. Determine whether the series is convergent, or divergent. Indicate the test/s you used a. 2n + 3 (b) ???.n! 4n2--3n35n

Explanation / Answer

1)
Using alt series test
bn = 5/ln(n)
which both goes to zero as n --> inf
and is decreasing
So, by alt series test, an = (-1)^n * 5/lnn CONVRGES

But does it converge absolutely?
Lets see

|an| = 5/ln(n)

We know that 5/ln(n) > 5/n

But 5/n diverges
So, by comparison test, something larger like 5/ln(n) has to diverge

So, the abs convergence aint happenin'

So, this CONVERGES CONDITIONALLY

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2)
Again by alt series test, it converges

But does |an| = 3/sqrt(n+4) converge?

We know that this diverges
becuase by integral test we get this to be infinity

So, C0NDITIONALLY CONVERGES

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3)
Ratio test
an = (-1)^n * n! / (n^3 * 2^(n+1))

a(n+1) = (-1)^(n+1) * (n+1)! / ((n+1)^3 * 2^(n+2))

Dividing,
a(n+1) / an becomes :

(-1)(n+1) * n^3/(n+1)^3 * 1/2

As n ---> inf
this goes to infinity too

So, DIVERGENT

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