Prove that series
n=1∑+∞an and n=m∑+∞an or converge together or diverge together. When they converge, find α to we have n=1∑+∞an=α+n=m∑+∞an
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I will prove that n=1∑+∞an converges ⇔ n=m∑+∞an converges.
We have the nth partial sums of seri n=1∑+∞an is sn=k=1∑nak and the nth partial sums of seri n=m∑+∞an is tn+m−1=k=m∑n+m−1ak (or the nth partial sums of seri n=m∑+∞an istn=k=m∑nak ? I think I have problem with this)
We prove that the sequence (sn) converges ⇔ sequence (tn+m−1) converges
First, suppose we have the sequence (sn) converges and n→∞limsn=b. We have ∀ϵ>0,∃n0∈N,∀n≥n0:∣sn−b∣<ϵ. Cause n+m−1≥n≥n0 so ∣sn+m−1−b∣<ϵ⇒∣tn+m−1+k=1∑m−1ak−b∣<ϵ.
Therefore ∀ϵ>0,∃n0∈N,∀n≥n0:∣tn+m−1−(b−k=1∑m−1ak)∣<ϵ. So (tn+m−1) converges and n→∞limtn+m−1=b−k=1∑m−1ak
Second, suppose we have the sequence (tn) converges and n→∞limtn=b. I'm stuck at this.
Please tell me where I was wrong. Thank you very much.
n=1∑+∞an and n=m∑+∞an or converge together or diverge together. When they converge, find α to we have n=1∑+∞an=α+n=m∑+∞an
----------------------
I will prove that n=1∑+∞an converges ⇔ n=m∑+∞an converges.
We have the nth partial sums of seri n=1∑+∞an is sn=k=1∑nak and the nth partial sums of seri n=m∑+∞an is tn+m−1=k=m∑n+m−1ak (or the nth partial sums of seri n=m∑+∞an istn=k=m∑nak ? I think I have problem with this)
We prove that the sequence (sn) converges ⇔ sequence (tn+m−1) converges
First, suppose we have the sequence (sn) converges and n→∞limsn=b. We have ∀ϵ>0,∃n0∈N,∀n≥n0:∣sn−b∣<ϵ. Cause n+m−1≥n≥n0 so ∣sn+m−1−b∣<ϵ⇒∣tn+m−1+k=1∑m−1ak−b∣<ϵ.
Therefore ∀ϵ>0,∃n0∈N,∀n≥n0:∣tn+m−1−(b−k=1∑m−1ak)∣<ϵ. So (tn+m−1) converges and n→∞limtn+m−1=b−k=1∑m−1ak
Second, suppose we have the sequence (tn) converges and n→∞limtn=b. I'm stuck at this.
Please tell me where I was wrong. Thank you very much.
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