How do you test the series #Sigma 1/(2^n-n)# from n is #[1,oo)# for convergence?
You can test the convergence of the series ( \sum_{n=1}^{\infty} \frac{1}{2^n - n} ) using the Ratio Test. Apply the Ratio Test by taking the limit as ( n ) approaches infinity of the absolute value of the ratio of the (n+1)-th term to the n-th term. If the limit is less than 1, the series converges; if it's greater than 1, the series diverges; if it's equal to 1, the test is inconclusive.
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The series converges by the ratio test.
We do a series ratio test
Here,
Therefore,
As,
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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.
- How do you determine whether the sequence #a_n=(2^n+3^n)/(2^n-3^n)# converges, if so how do you find the limit?
- How do you use the Ratio Test on the series #sum_(n=1)^oo(n!)/(100^n)# ?
- How do you find #lim lnt/(t-1)# as #t->1# using l'Hospital's Rule?
- How do you show that the harmonic series diverges?
- Prove that lim_(n->oo) (2n+1)/(3n+2)=2/3 ?

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