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lecture 19, 20 ,21

# lecture 19, 20 ,21 - MATH 118& 21 SERIES INTEGRAL&...

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Unformatted text preview: MATH 118, LECTUREs 19, 20 & 21: SERIES: INTEGRAL & COMPARISON TESTS 1 Series We are frequently interested in taking the sum of an infinite number of elements, that is to say, something of the form ∞ summationdisplay n =1 c n = c 1 + c 2 + c 3 + ··· + c n + ··· where the c i ’s are constant values. This is called an infinite series . We recall from a few weeks ago, we were interested in calculating the area under the curve of unbounded regions (improper integrals). Suppose we were asked to numerically integrate integraldisplay ∞ e- x dx using the rectangular rule and an interval width of h = 1. Unlike our typical examples, which had a bounded domain, we now have an infinite number of rectangles n which we need to sum up. This results in the estimation integraldisplay ∞ e- x dx ≈ ∞ summationdisplay n =1 e- n = e- 1 + e- 2 + e- 3 + ··· + e- n + ··· This approximation is pretty crude, and could be made better by using a small interval width h , or by using the trapezoidal of Simpson rules. We will put aside those concerns for the time being and focus on the nature of the sum. We are adding an infinite number of terms, all greater than zero—does this necessarily mean the sum is infinite? Before jumping to the exciting conclusion, let’s consider another way of looking at the series. It is clearly not possible to sum an infinite number of terms; however, we suspect we can get a general flavour of how the sum is 1 proceding by taking a number of sums of a finite number of elements. We call such sums partial sums and they are denoted by S n = n summationdisplay k =1 c k . For this sequence, we have S 1 = e- 1 ≈ . 36788 S 2 = e- 1 + e- 2 ≈ . 50321 S 3 = e- 1 + e- 2 + e- 3 ≈ . 55300 S 4 = ··· We notice that this generates the infinite sequence of partial sums { S 1 ,S 2 ,S 3 ,... ,S n ,... } . Furthermore, the question of whether the series is finite or infinite is precisely the question of whether the sequence of partial sums converges to some limit or not! We already have some sense of whether sequences converge or not based on various tests; unfortunately, they will prove of limited use in the discussion of the convergence of series. It is good, however, to have the intuition that series give rise to sequences of partial sums. We note that lim n →∞ S n = lim n →∞ parenleftBigg n summationdisplay k =1 c k parenrightBigg = ∞ summationdisplay n =1 c n . 1.1 Geometric Series We return to the question of whether the series ∑ ∞ n =1 e- n is finite or not (or, equivalently, whether the sequence of partial sums { S n } ∞ n =1 converges or not). This series fits the general form of a geometric series , which is ∞ summationdisplay n =1 ar n- 1 ....
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lecture 19, 20 ,21 - MATH 118& 21 SERIES INTEGRAL&...

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