How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given #1-2+3-4+...+n(-1)^(n-1)#?

Answer 1

To find the nth partial sum of the series (1 - 2 + 3 - 4 + \ldots + n(-1)^{n-1}), you need to analyze the pattern. The series alternates between positive and negative terms based on the power of (-1), with each term being the consecutive integer multiplied by (-1) raised to the power of one less than its position.

The nth partial sum (S_n) can be calculated using this pattern. If n is even, the sum will be negative, and if n is odd, the sum will be positive.

To determine whether the series converges, you can check if the sequence of nth partial sums approaches a limit as n approaches infinity. In this case, the series diverges because the terms do not approach a specific value but rather alternate between positive and negative integers.

Therefore, the series does not converge, and there is no finite sum when it exists.

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Answer 2

#sum_(n=1)^oo n(-1)^(n-1)#

is indeterminate.

The series does not converge as it does not satisfy the necessary condition for convergence:

#lim_(n->oo) a_n = 0#
We can evaluate the partial sums noting that for #n# even we can group the terms as:
#s_(2n) = (1-2)+(3-4)+...(2n-1-2n) =underbrace(-1-1+...-1)_(n " times") =-n#
while for #n# odd:
#s_(2n+1) = 1 + (-2 +3) + (-4+5)+...+(-2n+2n+1) = underbrace(1+1+1+...+1)_(n+1 " times") = n+1#

Hence the sums oscillate and are unbounded in absolute value so that:

#lim_(n->oo) s_n#

does not exist.

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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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