Notes on Series

Notes on Series - Notes on Series Definitions: Let cfw_ an...

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Notes on Series Definitions : Let { } 1 n n a = be a sequence. Define a new sequence { } 1 n n s = , called the sequence of partial sums for the sequence { } 1 n n a = , by the following equations: 1 1 s a = , 1 n n n s s a - = + for 2 n . This new sequence of partial sums is called the series of n a s, and it is denoted by 1 n n a = . For each 1 n , n s is called the th n partial sum for the series 1 n n a = . The number n a is called the th n term of the series (as well as the th n term of the sequence). There are several observations to keep in mind when dealing with series. These are listed below: Notice that the sequence of partial sums and the series of n a are mathematically the same thing. The term n s is the th n term of the sequence of partial sums { } 1 n n s = . If { } 1 n n s = is the same thing as the series 1 n n a = , one might ask why is n s not called the th n term of the series? The reason is, as we shall see later, that it is necessary to distinguish between n a and n s without specific reference to the sequence { } 1 n n a = . For 2 n , 1 n n n s s a - = + . This means 1 1 s a = 2 1 2 1 2 s s a a a = + = + 3 2 3 1 2 3 s s a a a a = + = + + 4 3 4 1 2 3 4 s s a a a a a = + = + + + M 1 1 2 3 n n n n s s a a a a a - = + = + + + + L So we often write 1 2 3 1 n n n a a a a a = = + + + + + L L Sometimes the index for the sequence { } 1 n n a = does not begin with the number 1. In such cases the corresponding series is also indexed accordingly. For example, if the sequence is { } 0 n n a = , then the series is 0 1 2 0 n n n a a a a a = = + + + + + L L ; while if the sequence is { } 3 n n a = , then the series is 3 4 5 3 n n n a a a a a = = + + + + + L L .
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When the index for the sequence { } n a does not matter in the discussion, we write n a , or, in the context of the sequence of partial sums, we write { } n s . Examples : Below are some examples that illustrate the difference between a sequence and the corresponding series and the sequence of partial sums. 1) Consider the constant sequence { } 1 n a = , where a is a constant. Observe that { } { } 1 , , , , , n a a a a a = = K K . The corresponding series is 1 n a a a a a = = + + + + + L L . Calculating the sequence of partial sums: 1 1 2 1 2 1 2 3 2 3 1 2 3 4 3 4 1 2 3 4 1 1 2 3 2 3 4 n n n n n times s a a s s a a a a a a s s a a a a a a a a s s a a a a a a a a a a s s a a a a a a a a a na - - = = = + = + = + = = + = + + = + + = = + = + + + = + + + = = + = + + + + = + + + + = M L L 1 4 42 4 43 So the sequence of partial sums is { } { } 1 ,2 ,3 , , , n an a a a an = = K K . 2)
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Notes on Series - Notes on Series Definitions: Let cfw_ an...

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