Sum of Series $\approx$ Error Estimation

In summary, the first 10 terms can be used to approximate the sum of the series $\sum_{n=1}^{\infty}\frac{1}{3^n+4^n}$, with an estimated error given by the integral $\int_{10}^{\infty} 1/4^x\; dx$. The upper bound on the error is provided by the first neglected term, $1/(3^{11}+4^{11})$.
  • #1
ineedhelpnow
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use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.

$\sum_{n=1}^{\infty}\frac{1}{3^n+4^n}$
 
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  • #2
ineedhelpnow said:
use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.

$\sum_{n=1}^{\infty}\frac{1}{3^n+4^n}$

I will leave you to sum the first ten terms, then we observe:

As $n$ becomes large $3^n \ll 4^n$ so $1/(3^n+4^4)\approx 1/4^n$ and $1/(3^n+4^4) < 1/4^n$ so the remainder after summing the first 10 terms of the series is approximately equal to (and less than) $\int_{10}^{\infty} 1/4^x\; dx$ ...

The above is an upper bound on the truncation error, in this case a lower bound is provided by the first neglected term: $1/(3^{11}+4^{11})$

.
 
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FAQ: Sum of Series $\approx$ Error Estimation

What is the sum of a series?

The sum of a series is the total of all the terms in the series. For example, in the series 1 + 2 + 3 + 4, the sum would be 10.

What is error estimation in relation to the sum of series?

Error estimation in relation to the sum of series is a method used to approximate the sum of a series by calculating the difference between the actual sum and an estimate. This can be helpful in cases where the actual sum is difficult to calculate or infinite.

How is error estimation calculated for the sum of a series?

Error estimation can be calculated using various methods, such as the difference between the actual sum and a partial sum, or by using a formula specific to the type of series. For example, for a geometric series, the error estimation formula is given by |Rn| ≤ |a| / (1 - |r|), where Rn is the remainder term, a is the first term, and r is the common ratio.

Why is error estimation important in mathematics?

Error estimation is important in mathematics because it allows us to approximate the value of a series without having to calculate the exact sum, which can be time-consuming or even impossible in some cases. It also helps us understand the accuracy and reliability of our calculations.

Can error estimation be used for all types of series?

No, error estimation methods may vary depending on the type of series. For example, the error estimation formula for an arithmetic series is different from that of a geometric series. Additionally, error estimation may not be applicable to all types of series, such as divergent series.

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