Answer to Question #184536 in Real Analysis for Leonard

Question #184536
  1. Use the definition of the limit  to show that the sequence (1 + (−1)^n) is divergent.
1
Expert's answer
2021-05-07T12:29:49-0400

"\\begin{array}{l}\n\\sum_{n=0}^{\\infty}\\left(1+(-1)^{n}\\right) \\rightarrow \\text { divergent } \\\\\n\\text { Rule: We have } \\sum_{k=0}^{\\infty} a_{k} \\text { , } \\\\\n\\text { if } \\lim _{k \\rightarrow \\infty} a_{k}=0 \\text { , the suries diverges } \\\\\n\\lim _{n \\rightarrow \\infty}\\left(1+(-1)^{n}\\right)=\\lim _{n \\rightarrow \\infty} 1+\\lim _{n \\rightarrow \\infty}(-1)^{n}= \\\\\n=1+1=2 \\\\\nn=2 k, k \\in N \\\\\n\\text { the series diverges }\n\\end{array}"



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