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### Course: AP®︎/College Calculus BC>Unit 10

Lesson 10: Alternating series error bound

# Worked example: alternating series remainder

Using the alternating series remainder to approximating the sum of an alternating series to a given error bound.

## Want to join the conversation?

• okay, @ why does R sub k start with "k+1"? Shouldn't it just start with k? I have been trying to figure this one out but it just doesn't seem to make sense for me.
• The k term is the last term of the partial sum that is calculated. That makes the k + 1 term the first term of the remainder. This is the term that is important when creating the bound for the remainder, as we know that the first term of the remainder is equal to or greater than the entire remainder. Sal discusses this property in the previous video.
• Why does the remainder need to be less or equal to 0.001? Where does this 0.001 come from?
• It's the question that Sal asked. He said in the video to determine the k that has its remainder less than or equal to 0.001.
• so what happens if the first term you neglect is negative do we just take the absolute value of that term?
• Yes, because we are interested in the "margin of error," so we want it to be within x (like 0.001 in the above video) of the original sum. It could either be 0.001 less than or more than S (it could go either way) and hence the sign of the term you neglect does not matter.
• In this unit, we learn a lot about different tests on series convergence/divergence. But sometimes I have no idea to use which test to apply on the problem. How can i solve it? thanks.
• It is possible |Rk| = |a(k+1)| ? or should I just say |Rk| just less and not equal to |a(k+1)| ?
• Yes it is possible. We're talking about alternating series that satisfy the requirements of the alternating series test, one of which is that the series is decreasing. Here "decreasing" only means "never increasing." It doesn't mean that successive terms can never be equal. If we get to a point where all the terms after a(k+1) are zero, then yes, |Rk| = |a(k+1)|.
(1 vote)
• I am not sure...how this entire unit of infinite series is related to Calculus?
(1 vote)
• It's related to calculus as soon as you start applying the limit process to an "infinite" series. This means that as soon as you say, "ok, I am gonna sum up the terms of this sequence from n=1 to n=infiniy," u r applying a limit process. U r not necessarily saying that the sum of the infinite series is a definite number, but u r saying that it approaches (calculus comes in) to some number. U do this using limits. Calculus is the mathematics of change and u can intuitively associate the sum of the series changing as n increases in an infinite series. Hope this helps! :)
• Why does R start with k+1 and not just k? Where does the +1 come from?
• Remember that 𝑆(𝑘) is the sum of the first 𝑘 terms, while 𝑅(𝑘) is the sum of the remaining terms.
This means that the last term of 𝑆(𝑘) is ±1∕√𝑘, and thereby the first term of 𝑅(𝑘) must be ∓1∕√(𝑘 + 1)
• How do I know whether Rk is less than or less than/equal to the value of the next term in the alternating series??