Question 11.11.1.8: Writing a Sum in Summation notation Express each sum using s......
Writing a Sum in Summation notation
Express each sum using summation notation.
(\mathbf{a})\;\;1^{2}+2^{2}+\;3^{2}\;+\cdot\cdot\cdot+9^{2}\qquad\qquad\qquad (\mathbf{b})\;\;1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdot\cdot\cdot+\frac{1}{2^{n-1}}(a) The sum 1^{2}\,+\,2^{2}\,+\,3^{2}\,+\,\cdot\,\cdot\,\cdot\,+\,9^{2}. has 9 terms, each of the form k^{2}, and starts at k = 1 and ends at k = 9:
1^{2}+2^{2}+3^{2}+\cdot\cdot\cdot\cdot\ +9^{2}=\sum_{k=1}^{9}k^{2}(b) The sum
1\,+\,{\frac{1}{2}}\,+\,{\frac{1}{4}}\,+\,{\frac{1}{8}}\,+\,\cdot\,\cdot\,\cdot\,+\,{\frac{1}{2^{n-1}}}
has n terms, each of the form {\frac{1}{2^{k – 1}}}, and starts at k = 1 and ends at k = n:
1\,+\,{\frac{1}{2}}\,+\,{\frac{1}{4}}\,+\,{\frac{1}{8}}\,+\,\cdot\,\cdot\,\cdot\,+\,{\frac{1}{2^{n-1}}}=\,\sum_{k=1}^{n}{\frac{1}{2^{k-1}}}Related Answered Questions
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