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For the series sum(n=1)(inf) (-1)^n*a_n where a_n = 1/n when n is even and 1/n^2 when n is odd, is it divergent?
For the series sum(n=1)(inf) (-1)^n*a_n where a_n = 1/n when n is even
In summary, the formula for the series sum(n=1)(inf) (-1)^n*a_n where a_n = 1/n when n is even is -1/2 + 1/4 - 1/6 + 1/8 - 1/10 + ... This series is calculated by plugging in the values of n (1, 2, 3, ...) into the formula and adding up the resulting terms. The value of this series is ln(2), or approximately 0.693147. It is convergent, meaning that the sum of the terms approaches a specific value as n gets larger. This series, known as the Alternating Harmonic Series, has significance in calculus, number theory
FAQ: For the series sum(n=1)(inf) (-1)^n*a_n where a_n = 1/n when n is even
What is the formula for the series sum(n=1)(inf) (-1)^n*a_n where a_n = 1/n when n is even?
The formula for this series is:
sum(n=1)(inf) (-1)^n*a_n = -1/2 + 1/4 - 1/6 + 1/8 - 1/10 + ...
How is this series calculated?
This series is calculated by plugging in the values of n (1, 2, 3, ...) into the formula and adding up the resulting terms. The series is said to converge because the values of the terms approach a specific value as n gets larger.
What is the value of this series?
The value of this series is ln(2), or approximately 0.693147.
Is this series convergent or divergent?
This series is convergent, meaning that the sum of the terms approaches a specific value as n gets larger.
What is the significance of this series in mathematics?
This series is known as the Alternating Harmonic Series and it plays an important role in calculus and number theory. It is also used in various proofs and applications in mathematics.