6. Consider the sequence {a} = {√² √2+√² √√2+√√2 + √² √2+√√2+√√2+√²-} n=1 Notice that this sequence can be recursively d

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6 Consider The Sequence A 2 2 2 2 2 2 N 1 Notice That This Sequence Can Be Recursively D 1 (27.91 KiB) Viewed 37 times

6. Consider the sequence {a} = {√² √2+√² √√2+√√2 + √² √2+√√2+√√2+√²-} n=1 Notice that this sequence can be recursively defined by a₁ = √2, and an+1 = √2+ an for all n> 1. (a) (5 points) Show that the above sequence is monotonically increasing. Hint: You can use induction. (b) (5 points) Show that the above sequence is bounded above by 3. Hint: You can use induction. (c) (2 points) Apply the Monotonic Sequence Theorem to show that lim, an exists. (d) (5 points) Find limnan (e) (3 points) Determine whether the series an is convergent. n=1