Estimate the error in using the partial sum of 65 terms of the series ∞∑k=1 9/k^2

Answer 1

By the Integral Test, the series converges.

Let S = sum of entire series and S₆₅ = sum of the first 65 terms.

Error = l S - S₆₅ l. Since the series has only positive terms, S > S₆₅, So, l S - S₆₅ l = S - S₆₅

S - S₆₅ < ∫_{(65 to ∞)} (9/x²)dx = lim_b→∞ (-9/x_{(65 to b)}) = lim_b→∞ (-9/b + 9/65) = 9/65

Answer 2

The error in using the partial sum of the series \( \sum_{k=1}^{\infty} \frac{9}{k^2} \) can be estimated using the remainder formula for an alternating series or a convergent series.

The remainder term \( R_n \) for the series \( \sum_{k=1}^{\infty} \frac{9}{k^2} \) after using the partial sum of \( n \) terms is given by:

\[ R_n = \left| \sum_{k=n+1}^{\infty} \frac{9}{k^2} \right| \]

In this case, we're using 65 terms, so we want to find \( R_{65} \). We can calculate this using the following Python code:

n_terms = 65

remainder = sum(9 / k**2 for k in range(n_terms + 1, 10**6 + 1))

print(f"The estimated error using the partial sum of 65 terms is approximately {remainder:.10f}")

This code calculates the remainder \( R_{65} \) by summing the terms from \( n+1 \) to a sufficiently large value (10^6 in this case). The result will give you an estimate of the error in using the partial sum of 65 terms of the given series.