How do you find partial sums of infinite series?

Answer 1

To find the partial sums of an infinite series, you add up a finite number of terms in the series. The partial sum of an infinite series up to the nth term is denoted by (S_n). The formula for finding the partial sum of an infinite series depends on the type of series.

For example:

  1. Arithmetic Series: If the series is an arithmetic series, where each term is obtained by adding a constant difference to the previous term, you can use the formula for the nth term of an arithmetic series: [ S_n = \frac{n}{2} \left(2a + (n-1)d\right) ] where (a) is the first term, (d) is the common difference, and (n) is the number of terms.

  2. Geometric Series: If the series is a geometric series, where each term is obtained by multiplying the previous term by a constant ratio, you can use the formula for the nth partial sum of a geometric series: [ S_n = a \frac{1 - r^n}{1 - r} ] where (a) is the first term, (r) is the common ratio, and (n) is the number of terms.

  3. Other Series: For other types of series, such as power series or alternating series, you may need to use different methods to find the partial sums. These methods could involve techniques like term-by-term summation, integration, or recursion.

In general, when finding partial sums of infinite series, it's important to consider convergence. If the series converges, then the partial sums will approach a finite limit as the number of terms increases. If the series diverges, the partial sums will not approach a finite limit. In such cases, the series may not have a sum.

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Answer 2

The sequence of partial sums of an infinite series is a sequence created by taking, in order: 1) the first term, 2) the sum of the first two terms, 3) the sum of the first three terms, etc..., forever and ever.

An infinite series is an expression of the form: #\sum_{k=1}^{\infty}a_{k}=a_{1}+a_{2}+a_{3}+a_{4}+\cdots# (this represents a "pretending" to add infinitely many numbers...even though such a thing is technically impossible to do). The infinite sequence #(a_{k})#, also written as #\{a_{k}\}# or #a_{1},a_{2},a_{3},\ldots#, is called the sequence of terms of the infinite series.
If, for each positive integer #n#, we let #s_{n}# be defined as the finite sum #s_{n}=\sum_{k=1}^{n}a_{k}=a_{1}+a_{2}+a_{3}+\cdots+a_{n}#, then the sequence #(s_{n})#, also written as #\{s_{n}\}# or #s_{1},s_{2},s_{3},\ldots#, is called the sequence of partial sums of the infinite series.
The two sequences #(a_{k})# and #(s_{n})# are related to each other, but they are different, and it's important to understand the relationships and differences.
It is the sequence #(s_{n})# that determines whether the original series #\sum_{k=1}^{\infty}a_{k}# converges or not ("sums" to a finite number or not). In fact, by definition , if #(s_{n})# converges to a number #s#, then the series #\sum_{k=1}^{\infty}a_{k}# is also said to converge to #s# and we write that it "equals" #s#: #\sum_{k=1}^{\infty}a_{k}=s# ( understand that this is just convenient notation...we are not literally adding up infinitely many numbers, though it usually doesn't hurt to pretend we are ).
On the other hand, if #(s_{n})# does not converge, if it diverges (in any way), then the series #\sum_{k=1}^{\infty}a_{k}# is also said to diverge.
Here are two important things to realize with regard to the sequence of terms #(a_{k})# and the sequence of partial sums #(s_{n})#:
1) If the sequence #(a_{k})# does not converge to zero (meaning it either converges to some other number or it diverges), then the sequence #(s_{n})# diverges, meaning that the series #\sum_{k=1}^{\infty}a_{k}# diverges.
2) If the sequence #(a_{k})# does converge to zero, then the sequence #(s_{n})# may or may not converge (so the series #\sum_{k=1}^{\infty}a_{k}# may or may not converge ). We need other tests to help us decide about convergence (a formula for the partial sums (if possible), ratio test, root test, comparison test, limit comparison test, alternating series test, absolute convergence test, etc...)

Since this answer is so long, I'll let you look in your textbook or online for examples to illustrate these ideas. You might start by using the following search terms: "geometric series", "p series", "harmonic series", "alternating harmonic series", as well as the tests mentioned in the previous paragraph.

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Answer 3

To find the partial sums of an infinite series, you add up a finite number of terms in the series. This involves selecting a specific number of terms to sum together, rather than attempting to sum the entire infinite series. The partial sum of an infinite series is denoted by ( S_n ), where ( n ) represents the number of terms being summed.

To find the partial sum of an infinite series, you can use one of several techniques, depending on the nature of the series. These techniques include:

  1. Arithmetic Series: If the series follows an arithmetic progression, you can use the formula for the sum of an arithmetic series, which is given by ( S_n = \frac{n}{2}(a_1 + a_n) ), where ( a_1 ) is the first term, ( a_n ) is the ( n )-th term, and ( n ) is the number of terms.

  2. Geometric Series: For a geometric series, you can use the formula for the sum of a geometric series, which is ( S_n = \frac{a_1(1 - r^n)}{1 - r} ), where ( a_1 ) is the first term, ( r ) is the common ratio, and ( n ) is the number of terms.

  3. Convergence Tests: If the series is convergent, you can use convergence tests such as the ratio test, root test, or comparison test to determine whether the series converges and find its sum.

  4. Taylor Series: Some infinite series can be represented as Taylor series expansions of functions. In such cases, you can use the Taylor series representation to find the partial sums.

Regardless of the technique used, finding the partial sums of infinite series often involves identifying a pattern or formula that allows you to compute the sum of a finite number of terms.

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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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