Calculus II help. Have a test coming?

I have used the Ratio test, and it is inconclusive, which no conclusion can be drawn about the convergence or divergence.

since it is Alternating, it is Alternating Series Test,

we have to try three rules:

( i )

it is alternating ====> true

( ii )

lim b_k = 0

k--> ∞

b_k = 1/√(k^2)

b_k = 1/k

lim (1/k) = (1/∞) = 0 but k^1, the exponent here is 1 which is divergent by p-series test since 1 ≤ 1

k--> ∞

let's try the third rule to see if convergent because if not, it will be divergent.

( iii )

b_(k + 1) ≤ b_k

1/(k + 1) ≤ 1/k when n = 1 ------> 1/2 ≤ 1 ------> true

1/(k + 1) ≤ 1/k when n = 2 ------> 1/3 ≤ 1/2 ------> true

1/(k + 1) ≤ 1/k when n = 3 ------> 1/4 ≤ 1/3

o we have check mark for the three rules, it is convergent by alternating series test, but since the p-series is divergent here(making a trouble for it), it is CONDITIONALLY CONVERGENT.

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It converges. In fact, it converges conditionally.

Observe that the function 1/√(k² + 6) is a sequence of positive real numbers that are monotonically decreasing to 0. Hence, by the alternating series test, ∑(k=1 to ∞) (-1)^k / √(k² + 6) is convergent.

To see why it only converges conditionally, observe that:

∑(k=1 to ∞) | (-1)^k / √(k² + 6) |

= ∑(k=1 to ∞) 1 / √(k² + 6),

which is divergent if you use the limit comparison test with the harmonic series = ∑(k=1 to ∞) 1 / k.