How do you show that the series #1/2+2/3+3/4+...+n/(n+1)+...# diverges?

Answer 1

Is divergent.

#1/n < n/(n+1) # and
#sum_(k=1)^oo 1/k < sum_(k=1)^oo k/(k+1)#
but #sum_(k=1)^oo 1/k# is the so called harmonic series which is divergent. So #sum_(k=1)^oo k/(k+1)# is also divergent.
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Answer 2

To show that the series ( \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \ldots + \frac{n}{n+1} + \ldots ) diverges, we can use the limit comparison test.

We can compare it to the harmonic series ( \sum_{n=1}^{\infty} \frac{1}{n} ).

Let's examine the sequence ( \frac{n}{n+1} ). Taking the limit as ( n ) approaches infinity:

[ \lim_{n \to \infty} \frac{n}{n+1} = 1 ]

Since this limit is nonzero and finite, we can conclude that ( \frac{n}{n+1} ) is equivalent to 1 as ( n ) approaches infinity.

Now, we compare the given series to the harmonic series:

[ \frac{1}{n+1} < \frac{1}{n} ]

Since ( \frac{1}{n+1} ) is less than ( \frac{1}{n} ), and the harmonic series diverges, by the limit comparison test, the given series also diverges.

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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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