How do you use the direct Comparison test on the infinite series #sum_(n=1)^oo5/(2n^2+4n+3)# ?

Answer 1

By making the denominator smaller,

#5/{2n^2+4n+2} le 5/n^2#.

Since

#sum_{n=1}^infty 5/n^2=5sum_{n=1}^infty1/n^2#
is a convergent p-series with #p=2>1#,
#sum_{n=1}^infty {5}/{2n^2+4n+2}#

converges by Comparison Test.

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Answer 2

To use the Direct Comparison Test on the infinite series ( \sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3} ), we compare it with another series whose convergence behavior is known.

  1. We need to find a series ( \sum_{n=1}^{\infty} b_n ) such that ( 0 \leq \frac{5}{2n^2 + 4n + 3} \leq b_n ) for all ( n ).
  2. If ( \sum_{n=1}^{\infty} b_n ) converges, and ( \frac{5}{2n^2 + 4n + 3} ) is bounded above by ( b_n ), then ( \sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3} ) also converges.
  3. If ( \sum_{n=1}^{\infty} b_n ) diverges, and ( \frac{5}{2n^2 + 4n + 3} ) is bounded below by ( b_n ), then ( \sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3} ) also diverges.

In this case, we notice that ( 2n^2 + 4n + 3 ) is always positive for ( n \geq 1 ). Thus, ( \frac{5}{2n^2 + 4n + 3} ) is always positive.

We can try to bound ( \frac{5}{2n^2 + 4n + 3} ) from above. Notice that for ( n \geq 1 ):

[ 2n^2 + 4n + 3 > 2n^2 ]

Hence:

[ \frac{5}{2n^2 + 4n + 3} < \frac{5}{2n^2} ]

Now, ( \sum_{n=1}^{\infty} \frac{5}{2n^2} ) is a p-series with ( p = 2 ), which is known to converge.

Therefore, by the Direct Comparison Test, since ( \frac{5}{2n^2 + 4n + 3} ) is bounded above by ( \frac{5}{2n^2} ) and ( \sum_{n=1}^{\infty} \frac{5}{2n^2} ) converges, ( \sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3} ) also converges.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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