How do you Determine whether an infinite sequence converges or diverges?

Answer 1
The sequence #{a_n}# converges if #lim_{n to infty}a_n# exists (having a finite value); otherwise, it diverges.

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Answer 2

To determine whether an infinite sequence converges or diverges, you can use various techniques such as:

  1. Limit Comparison Test: Compare the given sequence with a known sequence whose convergence or divergence is already established.

  2. Ratio Test: Check the limit of the absolute value of the ratio of successive terms in the sequence.

  3. Root Test: Evaluate the limit of the nth root of the absolute value of the nth term in the sequence.

  4. Integral Test: Compare the sequence with a corresponding integral to determine convergence or divergence.

  5. Direct Comparison Test: Compare the given sequence with another sequence whose behavior is known.

  6. Alternating Series Test: Assess convergence of alternating series by checking the terms satisfy certain conditions.

  7. Geometric Series Test: Determine convergence or divergence based on the common ratio of a geometric sequence.

  8. Divergence Test: If the limit of the sequence's terms does not approach zero, the sequence diverges.

Applying one or more of these tests based on the characteristics of the sequence will help determine its convergence or divergence.

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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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