How do you test for convergence for #1/((2n+1)!) #?

Answer 1

In the case you meant "test the convergence of the series : #sum_(n=1)^(oo)1/((2n+1)!)#"

the Answer is : it #color(blue)"converges"#

To find out, we can use the ratio test. That is, if #"U"_"n"# is the #n^"th"# term of this series
Then if, we show that #lim_(nrarr+oo)abs("U"_("n"+1)/"U"_n)<1# it means that the series converges
On the other if #lim_(nrarr+oo)abs(("U"_("n"+1))/"U"_n)>1# it means that the series diverges

In our case

#"U"_n=1/((2n+1)!)#
#" "# and
#"U"_("n"+1)=1/([2(n+1)+1]!)=1/([2n+3]!)#
Hence, #"U"_("n"+1)/"U"_n=1/((2n+3)!)÷1/((2n+1)!)=((2n+1)!)/((2n+3)!)#
#"Notice that" :# #(2n+3)! =(2n+3)xx(2n+2)xx(2n+1)!#
Just like : #10! =10xx9xx8!# We subtract #1# each time to get the next
So we have, #"U"_("n"+1)/"U"_n=((2n+1)!)/((2n+3)(2n+2)(2n+1)!)=1/((2n+3)(2n+2))#

Next we test,

#lim_(nrarr+oo)abs("U"_("n"+1)/"U"_n)#
#=lim_(nrarr+oo)abs(1/((2n+3)(2n+2)))=lim_(nrarr+oo)1/((4n^2+10n+6))=1/(+oo)=0" "# and #0# is less than #1#
Hence, it's quite safe to conclude that the series #color(blue)"converges" !#
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Answer 2

To test for convergence of the series ( \frac{1}{(2n+1)!} ), you can use the Ratio Test. The Ratio Test states that for a series ( \sum_{n=1}^{\infty} a_n ), if ( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 ), then the series converges absolutely.

For the given series, [ a_n = \frac{1}{(2n+1)!} ]

So, [ \frac{a_{n+1}}{a_n} = \frac{\frac{1}{(2(n+1)+1)!}}{\frac{1}{(2n+1)!}} = \frac{(2n+1)!}{(2(n+1)+1)!} = \frac{1}{(2n+3)(2n+2)(2n+1)!} ]

Taking the limit as ( n \to \infty ), we get:

[ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{1}{(2n+3)(2n+2)} = 0 ]

Since the limit is less than 1, the series ( \frac{1}{(2n+1)!} ) converges absolutely by the Ratio Test.

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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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